Page 1
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Number ndash Place value and Number lines
The position of each digit is important as it carries a specific value
654321
Tens
Units
Hundred Thousands
Ten Thousands
Thousands
Hundreds
1234
Tenths
Hundredths
Thousandths
Ten Thousandths
Line numbers up according to place
value when calculating
Each place to the left is 10 times larger
Number lines can be used to count up or down
1 2 3 4 5 6 7
Markings are the same distance apart each time
An increase means moving to the right a decrease means moving to the left
Useful for ordering numbers of different forms
minus4 minus2 0 2 4 6 8
35
10028minus3
1
2620
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Number ndash Rounding
Rules of rounding01234 56789 ROUND UPROUND DOWN
Place Value Significant FiguresDecimal Places
14672To the nearest ten
14670To the nearest hundred
14700To the nearest thousand
15000
125298To 1 decimal place
125To 2 decimal places
1253To 3 decimal places
12530
325484To 1 significant figure
300000
To 3 significant figures
325000Be careful when rounding up a nine (A double rounding will occur)
1 2 3 4
Count Right from the Decimal Point
To 2 significant figures
330000
Count Right from first non-zero Digit
1 2 3 4 5 6
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Number ndash Addition and Subtraction
Addition Subtraction
Chunking method
Column method
Addition and subtraction of positive and negative numbers
AddSubtract according to place value436 + 58
430 + 50 = 480
6 + 8 = 14
436 + 58 = 494
563 minus 237
563 minus 200 = 363
363 minus 30 = 333
333 minus 7 = 326
563 minus 237 = 326
Line up numbers according to place value436 + 58 563 minus 237
436058+
4941
563237minus
326
5 1
Carry over if column is greater than 10
Borrow from next column if too small
+ + =
+ minus =
minus minus =
minus + =
Filling the bath analogy
119860119889119889 ℎ119900119905 119908119886119905119890119903
119860119889119889 119888119900119897119889 119908119886119905119890119903
119877119890119898119900119907119890 119888119900119897119889 119908119886119905119890119903
119877119890119898119900119907119890 ℎ119900119905 119908119886119905119890119903
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Number ndash Multiplication and Division
Multiplication DivisionColumn method
BoxGrid method
Bus shelter method
Chinese
Tens in top half
Units in bottom half
Starting from the right
Add the diagonals Carry over onto
the next diagonal if you need to
523 times 76
523 times 76
Separate numbers
according to place value
Numbers first then zeros
Double check calculations
Add using column addition
523 times 76
Line up numbers
according to place value
Multiply by units first then tens etc
Donrsquot forget to add zeros
Add using column addition
Know times tablesRemainders carried over
Add zeros when numbers run out
Double decker division
958 divide 8
2016 divide 12 Split into factorsDivide by one factor
Divide answer by second factor
Negative numbers
Signs the same = Positive answer
Signs different = Negative answer
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Number ndash BIDMAS
Example 1
BracketsIndicesDivisionMultiplicationAdditionSubtraction
ORDER
Follow the order of BIDMAS to calculate
6 + 3 times 4
6 + 12
6 + 3 times 4Multiplication first
Addition second
18
3 + 8 times 4 + 6 divide 5 minus 2
3 + 8 times 10 divide 5 minus 2
3 + 16 minus 2
3 + 8 times 4 + 6 divide 5 minus 2Brackets first
Multiplication and Division second
Addition and Subtraction last
17
(9 minus 3 times 2) 2divide (10 divide 5)
(3) 2divide (2)
9 divide (2)
45
(9 minus 3 times 2) 2divide (10 divide 5)
Follow the order of operations
within brackets
Example 2
Example 3
=
==
=
=
==
=
=
=
=
If you can do nothing for a given step move on to the next
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Number ndash Prime Numbers Factors Multiples
Prime numbers MultiplesFactorsA Number is Prime if it has exactly 2 factors
1 and itself
The factors of a number are the numbers which
divide into it exactly
A number that features in the times table of
another number
1 is not a prime
number
2 is the only even prime
number
No other number can divide into it exactly
2 3 5 7 11 13 17 19 2329 31 37 41 43 47
No remainder when divided by a factor
Factor Pairs
Factors of 24
1 times 242 times 123 times 84 times 6
Prime numbers up to 50
1 2 3 4 6 8 12 24
The product of two integers will produce a multiple
Close link to factors
Times table knowledge important
Multiples of 7 between 30 and 60
7 14 21 28 35 42 49 56 63 70
Really useful for Prime Factor Decomposition HCF and LCM
1 is a factor of all whole
numbers
Multiples of 9
9 18 27 36 45 54 63 72 81 90hellip
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Number ndash Prime Factor Decomposition HCF LCM
Prime Factor Decomposition
HCFHighest common factor
Tree method and ladder method
LCMLowest common multiple
Break numbers down into prime factors
Divide by a prime
Multiply Primes
Write in index form
The largest number that divides into two or more numbers
Use long format of Prime Factor Decomposition
119867119862119865 119900119891 48 119886119899119889 120
2 times 2 times 3 = 12
119867119862119865 119900119891 84 119886119899119889 980
2 times 2 times 7 = 28
Prime factor decomposition
Identify shared factors
Multiply values
The smallest number that occurs in the times table of two
or more numbers
119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120
48 = 24 times 3180 = 22 times 32 times 5
24 times 32 times 5 = 720
Multiply together all prime factors apart
from duplicates
In index form MultiplyHighest Power of each
prime
6 = 2 times 345 3 times 3 times 5
2 times 3 times 3 times 5 = 90
=
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 2
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Number ndash Rounding
Rules of rounding01234 56789 ROUND UPROUND DOWN
Place Value Significant FiguresDecimal Places
14672To the nearest ten
14670To the nearest hundred
14700To the nearest thousand
15000
125298To 1 decimal place
125To 2 decimal places
1253To 3 decimal places
12530
325484To 1 significant figure
300000
To 3 significant figures
325000Be careful when rounding up a nine (A double rounding will occur)
1 2 3 4
Count Right from the Decimal Point
To 2 significant figures
330000
Count Right from first non-zero Digit
1 2 3 4 5 6
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Number ndash Addition and Subtraction
Addition Subtraction
Chunking method
Column method
Addition and subtraction of positive and negative numbers
AddSubtract according to place value436 + 58
430 + 50 = 480
6 + 8 = 14
436 + 58 = 494
563 minus 237
563 minus 200 = 363
363 minus 30 = 333
333 minus 7 = 326
563 minus 237 = 326
Line up numbers according to place value436 + 58 563 minus 237
436058+
4941
563237minus
326
5 1
Carry over if column is greater than 10
Borrow from next column if too small
+ + =
+ minus =
minus minus =
minus + =
Filling the bath analogy
119860119889119889 ℎ119900119905 119908119886119905119890119903
119860119889119889 119888119900119897119889 119908119886119905119890119903
119877119890119898119900119907119890 119888119900119897119889 119908119886119905119890119903
119877119890119898119900119907119890 ℎ119900119905 119908119886119905119890119903
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Number ndash Multiplication and Division
Multiplication DivisionColumn method
BoxGrid method
Bus shelter method
Chinese
Tens in top half
Units in bottom half
Starting from the right
Add the diagonals Carry over onto
the next diagonal if you need to
523 times 76
523 times 76
Separate numbers
according to place value
Numbers first then zeros
Double check calculations
Add using column addition
523 times 76
Line up numbers
according to place value
Multiply by units first then tens etc
Donrsquot forget to add zeros
Add using column addition
Know times tablesRemainders carried over
Add zeros when numbers run out
Double decker division
958 divide 8
2016 divide 12 Split into factorsDivide by one factor
Divide answer by second factor
Negative numbers
Signs the same = Positive answer
Signs different = Negative answer
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Number ndash BIDMAS
Example 1
BracketsIndicesDivisionMultiplicationAdditionSubtraction
ORDER
Follow the order of BIDMAS to calculate
6 + 3 times 4
6 + 12
6 + 3 times 4Multiplication first
Addition second
18
3 + 8 times 4 + 6 divide 5 minus 2
3 + 8 times 10 divide 5 minus 2
3 + 16 minus 2
3 + 8 times 4 + 6 divide 5 minus 2Brackets first
Multiplication and Division second
Addition and Subtraction last
17
(9 minus 3 times 2) 2divide (10 divide 5)
(3) 2divide (2)
9 divide (2)
45
(9 minus 3 times 2) 2divide (10 divide 5)
Follow the order of operations
within brackets
Example 2
Example 3
=
==
=
=
==
=
=
=
=
If you can do nothing for a given step move on to the next
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Number ndash Prime Numbers Factors Multiples
Prime numbers MultiplesFactorsA Number is Prime if it has exactly 2 factors
1 and itself
The factors of a number are the numbers which
divide into it exactly
A number that features in the times table of
another number
1 is not a prime
number
2 is the only even prime
number
No other number can divide into it exactly
2 3 5 7 11 13 17 19 2329 31 37 41 43 47
No remainder when divided by a factor
Factor Pairs
Factors of 24
1 times 242 times 123 times 84 times 6
Prime numbers up to 50
1 2 3 4 6 8 12 24
The product of two integers will produce a multiple
Close link to factors
Times table knowledge important
Multiples of 7 between 30 and 60
7 14 21 28 35 42 49 56 63 70
Really useful for Prime Factor Decomposition HCF and LCM
1 is a factor of all whole
numbers
Multiples of 9
9 18 27 36 45 54 63 72 81 90hellip
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Number ndash Prime Factor Decomposition HCF LCM
Prime Factor Decomposition
HCFHighest common factor
Tree method and ladder method
LCMLowest common multiple
Break numbers down into prime factors
Divide by a prime
Multiply Primes
Write in index form
The largest number that divides into two or more numbers
Use long format of Prime Factor Decomposition
119867119862119865 119900119891 48 119886119899119889 120
2 times 2 times 3 = 12
119867119862119865 119900119891 84 119886119899119889 980
2 times 2 times 7 = 28
Prime factor decomposition
Identify shared factors
Multiply values
The smallest number that occurs in the times table of two
or more numbers
119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120
48 = 24 times 3180 = 22 times 32 times 5
24 times 32 times 5 = 720
Multiply together all prime factors apart
from duplicates
In index form MultiplyHighest Power of each
prime
6 = 2 times 345 3 times 3 times 5
2 times 3 times 3 times 5 = 90
=
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 3
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Number ndash Addition and Subtraction
Addition Subtraction
Chunking method
Column method
Addition and subtraction of positive and negative numbers
AddSubtract according to place value436 + 58
430 + 50 = 480
6 + 8 = 14
436 + 58 = 494
563 minus 237
563 minus 200 = 363
363 minus 30 = 333
333 minus 7 = 326
563 minus 237 = 326
Line up numbers according to place value436 + 58 563 minus 237
436058+
4941
563237minus
326
5 1
Carry over if column is greater than 10
Borrow from next column if too small
+ + =
+ minus =
minus minus =
minus + =
Filling the bath analogy
119860119889119889 ℎ119900119905 119908119886119905119890119903
119860119889119889 119888119900119897119889 119908119886119905119890119903
119877119890119898119900119907119890 119888119900119897119889 119908119886119905119890119903
119877119890119898119900119907119890 ℎ119900119905 119908119886119905119890119903
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Number ndash Multiplication and Division
Multiplication DivisionColumn method
BoxGrid method
Bus shelter method
Chinese
Tens in top half
Units in bottom half
Starting from the right
Add the diagonals Carry over onto
the next diagonal if you need to
523 times 76
523 times 76
Separate numbers
according to place value
Numbers first then zeros
Double check calculations
Add using column addition
523 times 76
Line up numbers
according to place value
Multiply by units first then tens etc
Donrsquot forget to add zeros
Add using column addition
Know times tablesRemainders carried over
Add zeros when numbers run out
Double decker division
958 divide 8
2016 divide 12 Split into factorsDivide by one factor
Divide answer by second factor
Negative numbers
Signs the same = Positive answer
Signs different = Negative answer
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Number ndash BIDMAS
Example 1
BracketsIndicesDivisionMultiplicationAdditionSubtraction
ORDER
Follow the order of BIDMAS to calculate
6 + 3 times 4
6 + 12
6 + 3 times 4Multiplication first
Addition second
18
3 + 8 times 4 + 6 divide 5 minus 2
3 + 8 times 10 divide 5 minus 2
3 + 16 minus 2
3 + 8 times 4 + 6 divide 5 minus 2Brackets first
Multiplication and Division second
Addition and Subtraction last
17
(9 minus 3 times 2) 2divide (10 divide 5)
(3) 2divide (2)
9 divide (2)
45
(9 minus 3 times 2) 2divide (10 divide 5)
Follow the order of operations
within brackets
Example 2
Example 3
=
==
=
=
==
=
=
=
=
If you can do nothing for a given step move on to the next
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Number ndash Prime Numbers Factors Multiples
Prime numbers MultiplesFactorsA Number is Prime if it has exactly 2 factors
1 and itself
The factors of a number are the numbers which
divide into it exactly
A number that features in the times table of
another number
1 is not a prime
number
2 is the only even prime
number
No other number can divide into it exactly
2 3 5 7 11 13 17 19 2329 31 37 41 43 47
No remainder when divided by a factor
Factor Pairs
Factors of 24
1 times 242 times 123 times 84 times 6
Prime numbers up to 50
1 2 3 4 6 8 12 24
The product of two integers will produce a multiple
Close link to factors
Times table knowledge important
Multiples of 7 between 30 and 60
7 14 21 28 35 42 49 56 63 70
Really useful for Prime Factor Decomposition HCF and LCM
1 is a factor of all whole
numbers
Multiples of 9
9 18 27 36 45 54 63 72 81 90hellip
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Number ndash Prime Factor Decomposition HCF LCM
Prime Factor Decomposition
HCFHighest common factor
Tree method and ladder method
LCMLowest common multiple
Break numbers down into prime factors
Divide by a prime
Multiply Primes
Write in index form
The largest number that divides into two or more numbers
Use long format of Prime Factor Decomposition
119867119862119865 119900119891 48 119886119899119889 120
2 times 2 times 3 = 12
119867119862119865 119900119891 84 119886119899119889 980
2 times 2 times 7 = 28
Prime factor decomposition
Identify shared factors
Multiply values
The smallest number that occurs in the times table of two
or more numbers
119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120
48 = 24 times 3180 = 22 times 32 times 5
24 times 32 times 5 = 720
Multiply together all prime factors apart
from duplicates
In index form MultiplyHighest Power of each
prime
6 = 2 times 345 3 times 3 times 5
2 times 3 times 3 times 5 = 90
=
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 4
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Number ndash Multiplication and Division
Multiplication DivisionColumn method
BoxGrid method
Bus shelter method
Chinese
Tens in top half
Units in bottom half
Starting from the right
Add the diagonals Carry over onto
the next diagonal if you need to
523 times 76
523 times 76
Separate numbers
according to place value
Numbers first then zeros
Double check calculations
Add using column addition
523 times 76
Line up numbers
according to place value
Multiply by units first then tens etc
Donrsquot forget to add zeros
Add using column addition
Know times tablesRemainders carried over
Add zeros when numbers run out
Double decker division
958 divide 8
2016 divide 12 Split into factorsDivide by one factor
Divide answer by second factor
Negative numbers
Signs the same = Positive answer
Signs different = Negative answer
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Number ndash BIDMAS
Example 1
BracketsIndicesDivisionMultiplicationAdditionSubtraction
ORDER
Follow the order of BIDMAS to calculate
6 + 3 times 4
6 + 12
6 + 3 times 4Multiplication first
Addition second
18
3 + 8 times 4 + 6 divide 5 minus 2
3 + 8 times 10 divide 5 minus 2
3 + 16 minus 2
3 + 8 times 4 + 6 divide 5 minus 2Brackets first
Multiplication and Division second
Addition and Subtraction last
17
(9 minus 3 times 2) 2divide (10 divide 5)
(3) 2divide (2)
9 divide (2)
45
(9 minus 3 times 2) 2divide (10 divide 5)
Follow the order of operations
within brackets
Example 2
Example 3
=
==
=
=
==
=
=
=
=
If you can do nothing for a given step move on to the next
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Number ndash Prime Numbers Factors Multiples
Prime numbers MultiplesFactorsA Number is Prime if it has exactly 2 factors
1 and itself
The factors of a number are the numbers which
divide into it exactly
A number that features in the times table of
another number
1 is not a prime
number
2 is the only even prime
number
No other number can divide into it exactly
2 3 5 7 11 13 17 19 2329 31 37 41 43 47
No remainder when divided by a factor
Factor Pairs
Factors of 24
1 times 242 times 123 times 84 times 6
Prime numbers up to 50
1 2 3 4 6 8 12 24
The product of two integers will produce a multiple
Close link to factors
Times table knowledge important
Multiples of 7 between 30 and 60
7 14 21 28 35 42 49 56 63 70
Really useful for Prime Factor Decomposition HCF and LCM
1 is a factor of all whole
numbers
Multiples of 9
9 18 27 36 45 54 63 72 81 90hellip
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Number ndash Prime Factor Decomposition HCF LCM
Prime Factor Decomposition
HCFHighest common factor
Tree method and ladder method
LCMLowest common multiple
Break numbers down into prime factors
Divide by a prime
Multiply Primes
Write in index form
The largest number that divides into two or more numbers
Use long format of Prime Factor Decomposition
119867119862119865 119900119891 48 119886119899119889 120
2 times 2 times 3 = 12
119867119862119865 119900119891 84 119886119899119889 980
2 times 2 times 7 = 28
Prime factor decomposition
Identify shared factors
Multiply values
The smallest number that occurs in the times table of two
or more numbers
119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120
48 = 24 times 3180 = 22 times 32 times 5
24 times 32 times 5 = 720
Multiply together all prime factors apart
from duplicates
In index form MultiplyHighest Power of each
prime
6 = 2 times 345 3 times 3 times 5
2 times 3 times 3 times 5 = 90
=
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
copy EzyEducation ltd 2017
Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 5
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Number ndash BIDMAS
Example 1
BracketsIndicesDivisionMultiplicationAdditionSubtraction
ORDER
Follow the order of BIDMAS to calculate
6 + 3 times 4
6 + 12
6 + 3 times 4Multiplication first
Addition second
18
3 + 8 times 4 + 6 divide 5 minus 2
3 + 8 times 10 divide 5 minus 2
3 + 16 minus 2
3 + 8 times 4 + 6 divide 5 minus 2Brackets first
Multiplication and Division second
Addition and Subtraction last
17
(9 minus 3 times 2) 2divide (10 divide 5)
(3) 2divide (2)
9 divide (2)
45
(9 minus 3 times 2) 2divide (10 divide 5)
Follow the order of operations
within brackets
Example 2
Example 3
=
==
=
=
==
=
=
=
=
If you can do nothing for a given step move on to the next
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Number ndash Prime Numbers Factors Multiples
Prime numbers MultiplesFactorsA Number is Prime if it has exactly 2 factors
1 and itself
The factors of a number are the numbers which
divide into it exactly
A number that features in the times table of
another number
1 is not a prime
number
2 is the only even prime
number
No other number can divide into it exactly
2 3 5 7 11 13 17 19 2329 31 37 41 43 47
No remainder when divided by a factor
Factor Pairs
Factors of 24
1 times 242 times 123 times 84 times 6
Prime numbers up to 50
1 2 3 4 6 8 12 24
The product of two integers will produce a multiple
Close link to factors
Times table knowledge important
Multiples of 7 between 30 and 60
7 14 21 28 35 42 49 56 63 70
Really useful for Prime Factor Decomposition HCF and LCM
1 is a factor of all whole
numbers
Multiples of 9
9 18 27 36 45 54 63 72 81 90hellip
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Number ndash Prime Factor Decomposition HCF LCM
Prime Factor Decomposition
HCFHighest common factor
Tree method and ladder method
LCMLowest common multiple
Break numbers down into prime factors
Divide by a prime
Multiply Primes
Write in index form
The largest number that divides into two or more numbers
Use long format of Prime Factor Decomposition
119867119862119865 119900119891 48 119886119899119889 120
2 times 2 times 3 = 12
119867119862119865 119900119891 84 119886119899119889 980
2 times 2 times 7 = 28
Prime factor decomposition
Identify shared factors
Multiply values
The smallest number that occurs in the times table of two
or more numbers
119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120
48 = 24 times 3180 = 22 times 32 times 5
24 times 32 times 5 = 720
Multiply together all prime factors apart
from duplicates
In index form MultiplyHighest Power of each
prime
6 = 2 times 345 3 times 3 times 5
2 times 3 times 3 times 5 = 90
=
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 6
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Number ndash Prime Numbers Factors Multiples
Prime numbers MultiplesFactorsA Number is Prime if it has exactly 2 factors
1 and itself
The factors of a number are the numbers which
divide into it exactly
A number that features in the times table of
another number
1 is not a prime
number
2 is the only even prime
number
No other number can divide into it exactly
2 3 5 7 11 13 17 19 2329 31 37 41 43 47
No remainder when divided by a factor
Factor Pairs
Factors of 24
1 times 242 times 123 times 84 times 6
Prime numbers up to 50
1 2 3 4 6 8 12 24
The product of two integers will produce a multiple
Close link to factors
Times table knowledge important
Multiples of 7 between 30 and 60
7 14 21 28 35 42 49 56 63 70
Really useful for Prime Factor Decomposition HCF and LCM
1 is a factor of all whole
numbers
Multiples of 9
9 18 27 36 45 54 63 72 81 90hellip
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Number ndash Prime Factor Decomposition HCF LCM
Prime Factor Decomposition
HCFHighest common factor
Tree method and ladder method
LCMLowest common multiple
Break numbers down into prime factors
Divide by a prime
Multiply Primes
Write in index form
The largest number that divides into two or more numbers
Use long format of Prime Factor Decomposition
119867119862119865 119900119891 48 119886119899119889 120
2 times 2 times 3 = 12
119867119862119865 119900119891 84 119886119899119889 980
2 times 2 times 7 = 28
Prime factor decomposition
Identify shared factors
Multiply values
The smallest number that occurs in the times table of two
or more numbers
119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120
48 = 24 times 3180 = 22 times 32 times 5
24 times 32 times 5 = 720
Multiply together all prime factors apart
from duplicates
In index form MultiplyHighest Power of each
prime
6 = 2 times 345 3 times 3 times 5
2 times 3 times 3 times 5 = 90
=
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 7
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Number ndash Prime Factor Decomposition HCF LCM
Prime Factor Decomposition
HCFHighest common factor
Tree method and ladder method
LCMLowest common multiple
Break numbers down into prime factors
Divide by a prime
Multiply Primes
Write in index form
The largest number that divides into two or more numbers
Use long format of Prime Factor Decomposition
119867119862119865 119900119891 48 119886119899119889 120
2 times 2 times 3 = 12
119867119862119865 119900119891 84 119886119899119889 980
2 times 2 times 7 = 28
Prime factor decomposition
Identify shared factors
Multiply values
The smallest number that occurs in the times table of two
or more numbers
119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120
48 = 24 times 3180 = 22 times 32 times 5
24 times 32 times 5 = 720
Multiply together all prime factors apart
from duplicates
In index form MultiplyHighest Power of each
prime
6 = 2 times 345 3 times 3 times 5
2 times 3 times 3 times 5 = 90
=
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 8
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Number ndash Powers and Roots
Positive powers
Index notation used to write easily
Numbers multiplied by themselves
Negative powers Roots Fractional powers
4 times 4 times 4 = 43 119875119900119908119890119903
119861119886119904119890
84 = 8 times 8 times 8 times 8
2
3
2
=22
32=
2 times 2
3 times 3=
4
9
Anything to the power of 1 is just itself
51 = 5 281= 28
Know square numbers and cube numbers
Negative powers are fractions
A negative power means ldquoTake the reciprocal and make the
power positiverdquo
4
5
minus2
=5
4
2
=52
42=25
16Find Reciprocal
Apply Positive Power
Apply top and bottom
119909minus119899 ൗ1 119909119899
Roots are the inverse operations to powers
7 492
3119862119906119887119890 119903119900119900119905
4119865119900119906119903119905ℎ 119903119900119900119905
If you know square numbers and cube
numbers you can find their roots
2-Stage Powers
119909119898119899 = 119899 119909 119898
Numerator and denominator are important
Root by denominator first
Then power of numerator
1632 =
216
3= 64
27
64
minus23
=64
27
23
=4
3
2
=16
9
Negative Fractional PowersApply reciprocal first
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 9
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Algebra ndash Notation and Collecting like terms
Algebraic notation Collecting like terms
Algebra is the language we use to communicate mathematical information
Letters used to represent values are known as variables
Notation creates shortcuts
119886 times 119887 becomes 119886119887
119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102
coefficient
6119909119910 minus 5119886
119887+ 211199094
Expression
6119909119910 minus 5119886
119887+ 211199094
Terms
Same rules of BIDMAS applies to Algebra
Collecting like terms enables us to simplify expressions making them easier to use
Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified
Watch out for the sign before each term
5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910
5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910
21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909
Identify like terms
Use coefficients to collect like terms
First step in many problems involving Algebra
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 10
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Algebra ndash Formulae
Introduction Substitution Changing the subject
Explains how to calculate the value of a variable
ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is
charged a flat fee of pound2 and then pound3 for each mile drivenrdquo
119862 = 2 + 3119872For any given trip can easily work
out the cost of a taxi
119860 = 1205871199032
Area of circle formula
119878119906119887119895119890119888119905
Replace letters in the formula with numbers you are given
ldquoThe perimeter of a square is 4times the length of its sidesrdquo
119875 = 4119897What is perimeter of a
square with side length 5cm
119897 = 5 119875 = 4 5
119875 = 20119888119898Identify the formula and the values
to substitute in
Substitute values in using brackets
Carry out calculation remembering BIDMAS
Often it is useful to re-arrange a formula to make a different
variable the subject
119875 = 4119897
Make 119897 the subject of the formula
Use inverse operationsdivide 4divide 4
119875
4= 119897
119910 =18119905 minus 3
119901
Make 119905 the subject
119905 =119901119910 + 3
18
times 119901 +3 divide 18
Sometimes a variable will appear more than once in a formula
Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)
119886
5 + 119910= 119909Factorise first
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 11
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Algebra ndash Laws of indices
Basic Laws of Indices Advanced Laws of Indices
Special indices to consider
Anything to the power 1 = itself
Negative Indices
1199091 = 119909
1199090 = 1
1119909 = 1
Anything to the power 0 = 1
1 to the power of anything = 1
These laws can be applied if the bases are the same
119909119886 times 119909119887 = 119909119886+119887
119909119886 divide 119909119887 = 119909119886minus119887
119909119886 119887 = 119909119886times119887
When multiplying powers with the same base ndash Add the powers
When dividing powers with the same base ndash Subtract the powers
When raising the power (brackets) ndash Multiply the powers
1199113 times 1199117 = 11991110
1199042 divide 1199045 = 119904minus3
1198904 3 = 11989012
119909minus119899 ൗ1 119909119899Find Reciprocal
Apply Positive Power
Apply top and bottom
119911minus3 =1
119911
3
=1
11991136minus2 =
1
6
2
=1
36
Fractional Indices
119909119898119899 = 119899 119909 119898 Root by denominator first
Then power of numerator
Negative Fractional Indices
Negative Fractional PowersApply reciprocal first
119909minus119886119887 =
1119887 119909
119886
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 12
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Algebra ndash Expanding brackets
Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method
10 (119909 + 119910 + 4) = 10119909 +40+10119910
minus61199093119909 (6119909 minus 2) = 181199092
Expanding brackets often the first step in simplifying algebra
+15119909
F First terms
O Outside terms (119909 + 4)(119909 + 11)
I Inside terms
L Last terms +441199092
1199092
11119909
4119909
44
Multiply each term in first bracket by each term in second
Grid Method
(119909 + 4)(119909 minus 3) Split brackets up around grid
119909 +4119909
minus3
1199092 4119909
minus3119909 minus12 1199092 +119909 minus12
Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910
Include sign in multiplication
= minus12119909 + 13119910
Then simplify
Then simplify
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 13
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Algebra ndash Factorising
Highest common factor method Ladder method
The process where an expression has common factors removed and brackets introduced
119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911
12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911
181199092119910 + 6119909119910 minus 241199091199102119911
Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly
3(4119909 minus 2119910 + 119911)
HCF = 3 12119909 minus 6119910 + 31199113
4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)
HCF = 119886
119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)
119886
119909 + 119887119910 + 4119911
119886119909 + 119886119887119910 + 4119886119911
HCF = 6119909119910
6119909119910(3119909 + 1 minus 4119910119911)
Look at each term separately divide numbers first then the algebraic terms
181199092119910 + 6119909119910 minus 2411990911991021199112
91199092119910 + 3119909119910 minus 1211990911991021199113
31199092119910 + 1119909119910 minus 41199091199102119911119909
3119909119910 + 1119910 minus 41199102119911119910
3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 14
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Algebra ndash Linear Equations
Creating equations Advanced equations
A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0
Solving linear equations
Once the equation has been created it can be solved using a
balance method or inverse method
Expand brackets and simplify (collect like terms)
If 119909 is on both sides eliminate smallest value
Eliminate excess number
Divide and solve for 119909
General 4 step process
ORDER
3 119909 + 1 = 2(119909 + 2)
3119909+3 = 2119909+4
+3= 4119909
119909= 1
minus2119909minus2119909
minus3 minus3
Equations where fractions are involved
Fractions are divisions and can be eliminated by multiplying119909
2= 5
times 2times 2
119909 = 10
2119910
(3 minus 119910)= 4
times (3 minus 119910) times (3 minus 119910)
Remove variable from denominator
2119910 = 4 3 minus 119910
Cross-multiplying allows us to move terms in a fraction from one side of
an equation to the other
119909 + 1
3=119909
22 119909 + 1 = 3119909
3119896 minus 25119896
3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67
The perimeter of the triangle is
67cm Form an equation in 119909
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 15
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Algebra ndash Quadratics
An equation where the highest power of the variable is 2
1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics
119909 plusmn 119909 plusmnAim Convert quadratic into double brackets
Sum and product rule
1199092 + 119887119909 + 119888Add to make b
Multiply to make c
Establish Signs
If c is positive
1199092 + 5119909 + 6
Signs are same
(119909 + 3)(119909 + 2)
If c is negative Signs are different
1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example
1199092 minus 7119909 + 12
Positive c rarr Signs Same
Negative b rarr Both Minus
(119909 minus )(119909 minus )Factors of 12
12 times 16 times 2
4 times 3
Which pair make 7
(119909 minus 4)(119909 minus 3)
Factorising 119886 ne 1 Quadratics
119909 plusmn 119909 plusmnFactors of a to find
possible values= 5119909 plusmn 1119909 plusmn
51199092 minus 14119909 minus 3
3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =
6119909 plusmn 1119909 plusmnOR
Then find factors of c and see which satisfy b
Difference of Two Squares (DOTS)
1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)
1199092 minus 81 = (119909 + 9)(119909 minus 9)
41199102 minus 25 = (2119910 + 5)(2119910 minus 5)
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 16
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Algebra ndash Sequences
Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns
that follow a given ruleFinding the 119899119905ℎ term rule
Find the common difference (this will be your 119899 coefficient)
Write times table underneath sequence (of your 119899 coefficient)
Sequence minus times table (this is your extra bit)
Has the form 1198861198992 + 119887119899 + 119888A second layer difference
Halve 2nd layer difference for 1198992coefficient
Each number in the sequence is known as a lsquotermrsquo
Identify what is happening between each term to generate the rule
General rule is known as 119899119905ℎ term
Triangular Numbers
Square Numbers
Cube Numbers
1361015
1491625
182764125
Sequence 5 8 11 13
Times table 3 6 9 12
Extra bit +2 +2 +2 +2
+3 +3 +3
3119899 + 2
General formula
5 9 15 23 33 hellip+4 +6 +8 +10
+2 +2 +2
11198992 + 119887119899 + 119888Find linear sequence
Sequence 5 9 15 23
11198992 1 4 9 16
Subtract 4 5 6 7
119899119905ℎ term rule of this
= 119899 + 3
11198992 + 1119899 + 3
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 17
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Statistics ndash Mean Median Mode and Range
Mean Median Mode Range
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
Collect it all together and share it out evenly
Median = Middle value(Numbers written in order)
Using the mean to find the total amount
Use of formula to find location of median
Mode = Most common valueitem
Occurrence of no mode
Range = Largest - Smallest
Finds the middle valueAverage usually used for
qualitative dataReveals how closefar apart the values are
3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15
Mean =52
8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12
Interpreting measures of spread
119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =
119899 + 1
2Ezytown FC have scored an
average of 38 goals per game in their last 15 matches How many goals have they scored
38 times 15 = 57119892119900119886119897119904
The median of 45 values would be the 23rd number
when written in order45 + 1
2= 23
If every value appears equally there is no mode
1 1 3 3 7 7
Each value appears twice so there is no mode
The Smaller the range the closer and more lsquoconsistentrsquo
the values are
The Larger the range the more varied and more
lsquoinconsistentrsquo the values are
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 18
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Statistics ndash Averages from a frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
Cars Frequency 119891119909
0 4 0
1 11 11
2 12 24
3 7 21
4 6 24
Sum 40 80
This column is created by multiplying
the frequency (119891) by the number in
the category (119909)
This enables us to find out
the total amount which is
needed for the mean
times
times
times
times
times
Mean = 80
40= 2
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency
2119871119900119888119886119905119894119900119899 =119899 + 1
2
119871119900119888119886119905119894119900119899 =40 + 1
2= 205
The median lies between 20th and 21st value
Cars Frequency
0 4
1 11
2 12
3 7
4 6
Sum 40
4
15
27
Add down the frequency column When location
value has been exceeded that is the group where the
median lies
Median = 2
+
+
4 minus 0 = 4
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 19
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Weight Frequency
0119896119892 le 119909 le 2119896119892 12
2119896119892 lt 119909 le 4119896119892 3
4119896119892 lt 119909 le 6119896119892 9
6119896119892 lt 119909 le 8119896119892 10
Sum of Freq 34
Two extra columns are created the midpoint of each group and the 119891119909
column
Weight Frequency Mid-point 119891119909
0119896119892 le 119909 le 2119896119892 12 1 12
2119896119892 lt 119909 le 4119896119892 3 3 9
4119896119892 lt 119909 le 6119896119892 9 5 45
6119896119892 lt 119909 le 8119896119892 10 7 70
Sum of Freq 34 Sum of 119891119909 136
Statistics ndash Averages from a grouped frequency table
Mean
Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904
119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904
MedianMedian = Middle value
(Numbers written in order)
ModeMode = Most common
valueitem
Range
Range = Largest - Smallest
times
times
times
times
Mean = 136
34= 4119896119892
Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899
119879119900119905119886119897 119891119903119890119902119906119890119899119888119910
The category with the highest frequency119871119900119888119886119905119894119900119899 =
119899 + 1
2
119871119900119888119886119905119894119900119899 =34 + 1
2= 175
The median lies between 17th and 18th value
12
15
22
Add down the frequency column
When location value has been exceeded
that is the group where the median
lies
Median class =
+
+
8 minus 0 = 8119896119892
This enables us to find
out an estimate for
the mean
0119896119892 le 119909 le 2119896119892
4119896119892 lt 119909 le 6119896119892
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 20
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Statistics ndash Representing data
Bar Charts
Pictograms
Pie Charts
Line Charts
Title Axes
Scales Labels
Bars (equal width) (equal gaps)
Read off barsEllie = 40 hours
Nationality Guests
Spanish 30
British 24
French 10
German 8
Total 72
Find degrees per value 360deg divide 119905119900119905119886119897
1 119866119906119890119904119905 =360deg
72= 5deg
Angle
150deg
120deg
50deg
40deg
Farmer Pumpkins
Harry 20
Sami 50
Doug 40
Rachael 25
Using a symbol to represent certain amount Line charts are useful for displaying time series data
Data points within the lines are important
Lines visualise change
Extract required data carefully
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 21
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Statistics ndash Cumulative Frequency tables and graphs
Cumulative Frequency tables Cumulative Frequency graphs
Running total (add up as we go down)
Time Frequency Cumulative Frequency
45 le 119909 lt 48 3 3
48 le 119909 lt 50 8 11
50 le 119909 lt 52 16 27
52 le 119909 lt 55 12 39
55 le 119909 lt 60 11 50
60 le 119909 lt 70 2 52
The difference between cumulative frequency values will tell you the frequency
Points Cumulative Frequency
0 ndash 4 6
5 ndash 9 18
10 ndash 15 25
16 ndash 20 30
0
5
10
15
20
25
30
0 5 10 15 20C
um
ula
tive
Fre
qu
ency
Points Scored
Always plot each point at the end of
the group
Draw a smooth line through the points
How many games did the player
score more than 12 points22
30 minus 22 = 8 119892119886119898119890119904
May be asked to calculate percentages
=119860119898119900119906119899119905
119879119900119905119886119897times 100
What percentage of games does the player score less than 12 points
22
30times 100 = 73
Equals
Equals
Equals
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 22
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Statistics ndash Quartiles and box plots
Quartiles Box plotsBox plots allow us to visualise the spread of
a datasetInterested in specific points along the distribution
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100
Cu
mu
lati
ve F
req
uen
cy
Marks
Upper Quartile = 75
Median = 50
Lower Quartile = 25
The IQR provides a measure of the spread of the middle 50 of
the data
0 10 20 30 40 50 60 70 80 90
Exam Marks
Minimum Value
Maximum Value
Q1
Also used to compare two or more different datasets
0 10 20 30 40 50 60 70 80 90
Look to compare medians IQR and range
Q3
Q2
It is less affected by outliers than the range
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 23
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Age Frequency Class Width FD
20 le 119909 lt 25 300 5 60
25 le 119909 lt 30 150 5 30
30 le 119909 lt 40 100 10 10
40 le 119909 lt 60 100 20 5
Statistics ndash Histograms
A special type of bar chart for grouped data
Create the Frequency
density column
Frequency Density on the vertical axis
Bars can be different widths depending on the group size
119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910
119862119897119886119904119904 119882119894119889119905ℎ
Calculating Frequency from Histograms
dividedividedividedivide
119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ
Age FD
0 lt 119909 le 10 3
10 lt 119909 le 25 4
25 lt 119909 le 30 5
30 lt 119909 le 40 3
40 lt 119909 le 50 25
Class Width Frequency
10 30
15 60
5 25
10 30
10 25
timestimestimestimestimes
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 24
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Statistics ndash Scatter Graphs
We use scatter graphs when we are interested in the relationship between two variables
Points plotted like coordinates
Labour Party
Year Vote () Seats
2015 31 232
2010 29 258
2005 35 356
2001 41 413
1997 43 418
1992 34 271200
250
300
350
400
450
25 30 35 40 45
Labour Party Electoral Performance
Vote ()
Seat
s w
on
As one value increases so does the other
As one value increases the other decreases
No link between the two variables
One of the main incentives for drawing lines of best fit is to make predictions
Place line of best fit through the middle of the data
(Ignore Outliers)
Predict values by reading off the line
40 = 390 119904119890119886119905119904
Interpolation
Extrapolation
Predictions made within the dataset
Predictions made outsideof the dataset
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 25
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Number ndash Fractions ndash Simplifying Improper Mixed
Simplifying Improper Fractions Mixed Number
Divide both the numerator and denominator by the same value
Turning into a mixed number
The combination of a WHOLE number and a Fraction
Repeat the process until the fraction is in its simplest form
Even = divide 2
Odd = divide 357hellip
The numerator is larger than the denominator
1199033
Divide numerator by denominatorto get whole number
Remainder forms new numerator
Denominator remains the same
Turning into an improper fraction
Multiply whole number by denominator
Add on the numerator56 + 3
Denominator remains the same59
8Donrsquot forget to simplify your answers where necessary
Useful skills for addingsubtractingmultiplying and dividing fractions
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 26
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Number ndash Fractions ndash Addition and Subtraction
AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a
common denominator
2
9+5
9
7
9
When denominators are the same simply add the numerators
+
7
9minus
1
6When denominators are different
multiply the fractions
times 2 times 3
14
18minus
3
18
11
18
minus
Remember to simplify your answers
Method 1 ndash Deal with whole numbers and fractions separately
31
2+ 4
1
43 + 4 +
1
2+1
4= 7
3
4
52
3minus 2
1
95 minus 2 +
2
3minus1
9= 3
5
9
Method 2 ndash Convert to improper fractions first then calculate
61
5minus 4
3
4
31
5minus19
4
124
20minus95
20
29
20= 1
9
20
31
5+ 5
9
10
16
5+59
10
32
10+59
10
91
10= 9
1
10
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 27
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Number ndash Fractions ndash Multiplying and Dividing
Multiplying
Multiply across the top and bottom
Dividing
Check to see if you can cross cancel
5
1
2
1
10
3divide
2
3
10
3times
3
2
Multiply by the Reciprocal
Finding the reciprocal of a fraction swaps the numerator and denominator
10
21
21
5divide 3
3
10
1
31
2
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 28
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50100
Number ndash Fractions ndash Converting Decimal to Fraction
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Using place value
0123403
01 ൗ1 10
ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10
Write 032 as a fraction in its simplest form
03225
32 divide 2
divide 2
16divide 2
divide 2
8Know your place values
Place Decimal part over 101001000 etc
Simplify the Fraction
Put back whole numbers if you had them
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 29
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Number ndash Fractions ndash Converting Fraction to Decimal
Decimal -gt Fraction Conversions you should know
Decimal Fraction
05 ൗ1 2
0 ሶ3 ൗ1 3
025 ൗ1 4
02 ൗ1 5
0125 ൗ1 8
01 ൗ1 10
Equivalent Fraction method
075
025 ൗ1 4
025 + 025 + 025ൗ3 4
Multiply or Divide the fractions so that they can be converted to decimals easily
Division methodDivide the numerator by the denominator
Using Bus shelter division
13
20times 5
100
65= 065
Mixed Numbers and Improper Fractions
Process does not change
Try and work with mixed numbers where possible
344
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 30
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Number ndash Fractions ndash Converting recurring decimals
More complex recurring decimalsBe in a position to eliminate the Decimal numbers
020 ሶ5= 119909(times 100) (times 100)020 ሶ5
Move recurring decimal up to the decimal point
20 ሶ5100119909 =(times 10) (times 10)
205 ሶ51000119909 =
Be in a position to eliminate the
Decimal numbers
119909 = 185900
=37
180
Move recurring decimal up to the decimal point by multiplying by 10100 etc
Create equation by labelling the decimal 119909
Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc
Subtract two equations to eliminate recurring decimals and convert into fraction
Simplify your answer
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 31
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RPR ndash Quantities as fractionspercentages of each other
Quantities as fractions of each other Quantities as percentages of each other
Think of it as a test score
Fractions of amounts
Percentages of amounts
Express 5 as a fraction of 25
5
25
1
5
Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their
earnings
Nigel =30
90=
1
3Sanjay =
35
100=
7
20
0 ሶ3 035
Divide amount by denominator
Then multiply by the numerator
3
5of 60 60 divide 5 = 12 12 times 3 = 36
Express 5 as a percentage of 20
Convert to fraction or decimal then to percentage
5
20
divide 5
divide 5
25
100
times 5
times 525
Equivalent fraction over
100
Method 1
Convert to decimal
Method 2
025 255 divide 20times 100
times 100
Method 1- Unitary method
Find 1 10 5 etc
Method 2- Decimal method
Turn to a decimal divide 100 Then multiply by
amount
Find 35 of 40
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 32
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RPR ndash Percentages and percentage change
Introduction to percentages Percentage increasedecrease
Calculate percentage of amountAdd on for increase Subtract for decrease
Means out of 100
The common
conversions you should really know
Conversion Flow Chart
Increase 70 by 15
Method 1- Unitary method 10 = 7 5 = 35 15 = 105
Method 2- Decimal method 15 = 015 015 times 70 15 = 105
119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890
119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100
times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903
divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903
OriginalValue
New Value
Method 3- Calculator method
Increase or Decrease
Add to 100 for increase
Subtract from 100 for decrease
Convert to decimal (divide 100)
This is your multiplier factor
119868119899119888119903119890119886119904119890 119900119891 23 = times 123
119863119890119888119903119890119886119904119890 119900119891 42 = times 058
To find original amount work backwards and divide
by multiplier factor
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 33
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RPR ndash Simple interest and Compound Growth and Decay
Simple Interest Compound Growth
Follows the I=PRY formula
Interest RatePrincipal Yearstimes times
Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years
pound350times009times4= pound126Interest
Remember to convert the to a decimal
To calculate other parts of the formula you will need to change the subject
How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate
of 55 pa
Interest
RatePrincipal
Years
times
An amount is increased or decreased by a percentage
The process is repeated several times at each interval
The most efficient way to do this is using a Multiplier
Interest 1 plusmn RatePrincipal
Years
times
Growth
Decay
The general formula for compound
growth and decay
pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years
pound4000 times 1053= pound463050
A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years
pound15000times0854= pound783009
To calculate other parts of the formula you will need to change the subject
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 34
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RPR ndash Ratio
Introduction Sharing in a given ratio Map scale factors
Is the relationship between two or more quantities
It is written in the form 119938 ∶ 119939
Compares one part to another part
The ratio of red to blue is 4 ∶ 5
The ratio of blue to red is 5 ∶ 4
Sentence structure is important
Simplifying Ratios
12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2
Divide all numbers by the same valueThe ratio of boys to girls in a
Geography class is 15 ∶ 5What fraction of the class is girls
5
20
119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904
1
4
Share $40 in the ratio 3 ∶ 5Find total
number of partsAdd the ratio
parts together
3 + 5 = 8Find value of
one partDivide amount by number of parts
$40 divide8 = $5Each part of the ratio is worth $5
Multiply by original ratio
3 ∶ 5times $5
$15 ∶ $25
Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have
27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905
4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)
It is the ratio of a distance on the mapmodel to the corresponding
size in real life
Written in the form 120783 ∶ 119951
Map or
Model
times 119904119888119886119897119890 119891119886119888119905119900119903 Real life
divide 119904119888119886119897119890 119891119886119888119905119900119903
Know your conversions
10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898
A map has a scale of 125000 Michael is 6cm from his home
How far from home is he Give your answer in 119896119898
6119888119898 times 25000 150000119888119898=
150000119888119898 1500119898 15119896119898divide 100 divide 1000
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 35
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RPR ndash Proportion
Direct Proportion Inverse ProportionDirect Proportion
As one value increases the other increases at the same rate
Inverse Proportion
As one value increases the other decreases at the same rate
Three Coffees cost pound750 How much would five Coffees cost
Find the value of one coffee then multiply by quantity needed
pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890
pound250 times 5 = pound1250
It takes 3 men 4 days to build a wall How long would it take 2 men
Find the time taken by one man then divide by quantity stated
3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904
12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904
119910 is directly proportional to 119909
119910 prop 119909
119910 = 119896 times 119909 119896 is the rate of change
Constant of proportionality
Work out the value of 119896
Compare two values
Form equation to solve problems
Solving direct proportion problems119901 is directly proportional to 119905
119901 = 24 119905 = 8a) Find 119901 when 119905 = 7
b) Find 119905 when 119901 = 39
119901 = 119896 times 119905 24 = 119896 times 8
24
8= 119896 3 = 11989624 = 119896 times 8
divide 8
119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21
119887) 39 = 3 times 119905divide 3
119905 = 13
119910 is inversely proportional to 119909
119910 prop1
119909119910 =
119896
119909
Constant of proportionality
119896
Solving inverse proportion problems119901 is inversely proportional to 119905
119901 = 16 119905 = 2a) Find 119901 when 119905 = 8
b) Find 119905 when 119901 = 64Compare two values
119901 =119896
11990516 =
119896
2
Work out the value of 119896times 2
32 = 119896
Form equation to solve problems
119901 =32
119905119886) 119901 =
32
8= 4
119887) 64 =32
119905119905 =
32
64= 05
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 36
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RPR ndash Graphical representations of Proportion
Direct Proportion Inverse Proportion
119910 prop 119909 119910 = 119896 times 119909As one value increases so
does the other
Linear relationships
Non - linear relationships
119910 prop1
119909119910 =
119896
119909
As one value increases the other decreases
Inverse proportion will always give a Curved graph
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 37
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Geometry ndash Quadrilaterals
Know the names of these Quadrilaterals and their properties
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 38
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Geometry ndash Triangles
Know the names of these Triangles and their properties
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 39
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Geometry ndash Polygons
RegularSide lengths are the
same
Interior and exterior angles the same
IrregularSide lengths are not
ALL the same
Interior and exterior angles the not ALL the
same
Equal sides are marked with a dash through the line
ConvexThe shape is lsquobulgingrsquo
outwards
All angles less than 180deg
ConcaveThe shape has lsquocavedrsquo
inwards
One or more angles is greater than 180deg
They are classified by the number of sides they have
Number of sides Name of shape
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 40
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Geometry ndash 3D shapes
The same cross sectional area throughoutPrisms
Faces
Faces Edges and Vertices
The flat surface of a 3-D shape
Edges
Vertices
Where two Faces meet
Where two or more edges meet at a point
Shape Faces Edges Vertices
Cube 6 12 8
Cuboid 6 12 8
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Square based Pyramid 5 8 5
Cone 2 1 1
Cylinder 3 2 0
Sphere 1 0 0
Faces + Vertices ndash Edges = 2
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 41
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Geometry ndash Angle facts
Introduction
A straight line Around a point Vertically opposite
A measure of turn Using a protractor
It has the unit degrees (deg)Reflex
Obtuse
Right
Acute Straight
Full turn
Greater than 0deg less than 90deg
Exactly 90deg
Greater than 90deg less than 180deg
Exactly 180deg
Greater than 180deg less than 360deg
Exactly 360deg
Has a square in the angle to indicate that it is 90deg
Looks like a book closing or crocodile jaws
Looks like a book falling open
A half turn to create a straight line
The larger angle outside the acute or obtuse angle
A movement around a point to create a circle
All angles on a straight line will add up to make 180deg
All angles around a point will add up to make 360deg
Where two straight lines cross opposite angles are
equal
30deg150deg
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 42
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Geometry ndash Angles in triangles and polygons
Triangles Polygons
All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg
Calculate what you
already know
149deg180deg minus
31deg119909 =
112deg37deg
149deg
+Subtract
from 180deg
Exterior angle = Sum of two angles on opposite side
107deg55deg 119909deg=+
162deg 119909deg=
Isosceles triangle
It has two equal lengths
It has two equal angles
Knowledge of triangles is important
Number of sides
Number of
Sum of interior angles
Regular interior angle
Regular exterior
angle
3 1 180deg 60deg 120deg
4 2 360deg 90deg 90deg
5 3 540deg 108deg 72deg
6 4 720deg 120deg 60deg
7 5 900deg 129deg 51deg
8 6 1080deg 135deg 45deg
119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg
119899360deg divide 119899
The number of triangles in a shape will always be TWO less than the number of sides
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 43
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Geometry ndash Angles Parallel lines
Alternate Corresponding Co-Interior
Two lines that travel side by side keeping the same distance apart at all times never intersecting
When a line intersects the parallel lines
Eight angles are created
The angles on one parallel
line match up with the angles
on the other parallel line
One Acute angleOne Obtuse angle
Different lines will
create different sets
of angles
Alternate angles are the same
lsquoZrsquo shape lsquoFrsquo shape
Corresponding angles are the same
lsquoCrsquo shape
Co-interior angles make 180deg
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 44
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Geometry ndash Pythagorasrsquo Theorem
1198882 = 1198862 + 1198872
For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other
two sides
Finding the Hypotenuse Finding the Shorter side
If you know the lengths of the two shorter sides you can calculate
the length of the hypotenuse
Square the two sides
Add them
Square root for answer
8
6
1198882 = 82 + 62
1198882 = 64 + 36
1198882 = 100119862 = 10
( )
If you know the Hypotenuse and a shorter side you can calculate the
length of the other shorter side
Square the two sidesSubtract
themSquare root for answer
119886
6
1198882 minus 1198872 = 1198862
122 minus 62 = 1198862144 minus 36 = 1198862
108 = 1198862
1039 = 119886( )
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 45
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Geometry ndash Trigonometry functions
Sine Function Cosine Function Tangent Function
119904119894119899120579 Value
0deg 0
30deg 05
45deg 2
2= 0707
60deg 3
2=0866
90deg 1
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
To get sinminus1
you have to press shift first
Work out side lengths and angles when given the opposite and
hypotenuse
Only used to calculate an angle
Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent
The angle is often described as theta which
is the Greek letter (120579)
Work out side lengths and angles when given the adjacent and
hypotenuse
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
To get cosminus1
you have to press shift first
Only used to calculate an angle
119888119900119904120579 Value
0deg 1
30deg 3
2= 0866
45deg 2
2= 0707
60deg 05
90deg 0
Work out side lengths and angles when given the adjacent and
opposite
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
119905119886119899120579 Value
0deg 0
30deg 3
3
45deg 1
60deg 3
To get cosminus1
you have to press shift first
Only used to calculate an angle
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 46
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Geometry ndash SohCahToa
Decide which ratio to apply
Label the triangle accurately with OAH
Use the triangles to help
When to use trigonometry
Find a missing side when given a length and an angle
Find an angle when given two lengths
119952119955
Calculating a missing side Calculating a missing angle
42deg
14
119909119874
119867
119860119914 119919
119912
Cover up what youneed to find
119909 = 14 divide cos(42deg)= 1884
119931 119912
119926
Cover up what youneed to find
119909 = tan 64deg times 115= 2358
Cover up what youneed to find
119930 119919
119926
119909 is an angle so use inverse
119909 = sinminus1(14 divide 378)
= 217deg
Use the functions and substitute
Alternate process
119904119894119899120579 =119900119901119901119900119904119894119905119890
ℎ119910119901119900119905119890119899119906119904119890
119888119900119904120579 =119886119889119895119886119888119890119899119905
ℎ119910119901119900119905119890119899119906119904119890
119905119886119899120579 =119900119901119901119900119904119894119905119890
119886119889119895119886119888119890119899119905
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 47
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Graphs ndash Coordinates
A set of values that indicate the position of a point
They normally occur in pairs in the form (119909 119910)
(119909 119910)
Direction along the 119909 - axis
Direction updownthe 119910 - axis
Along the corridor
Up Down the stairs
Start from a central point(00) - Origin
Reading the coordinates will lead you to the exact position
(7 minus4) Seven units rightFour units down
(minus26) Two units left Six units up
(minus5minus2) Five units left Two units Down
119910
119909
(minus54)
(77)
(minus3minus4)
(9 minus8)
119874119903119894119892119894119899
Four Quadrants Plotting coordinates
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 48
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Graphs ndash Equation of a Straight line
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119962 = 120790
119962 = 120785
119962 = minus120787
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
119961 = minus120791 119961 = 120786
Horizontal lines 119910 =
Vertical lines 119909 =
All straight line graphs follow the same rule
119910 = 119898119909 + 119888Gradient 119910 intercept
119862ℎ119886119899119892119890 119894119899 119910
119862ℎ119886119899119892119890 119894119899 119909
119877119894119904119890 119906119901
119877119906119899 119886119897119900119899119892
Gradient is the lsquosteepnessrsquo of the line
Calculated by or
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4 5
Equation of line from coordinates
Calculate gradient between points (119898)
Substitute in points and solve (119888)
Find the equation of the line that passes through (02) and (38)
Gradient=1199102minus1199101
1199092minus1199091
6
3= 2 (119898)
119910 = 2119909 + 119888 8 = 2(3) + 119888
8 = 6 + 119888 2 = 119888
119910 = 2119909 + 2
119904119906119887119904119905119894119905119906119905119890
119904119900119897119907119890
119910 = 119898119909 + 119888
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 49
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Graphs ndash Midpoints Parallel lines and Perpendicular lines
Midpoints Parallel lines Perpendicular lines
A midpoint is the halfway point between two end points of a line
segment119909119860 + 119909119861
2119910119860 + 119910119861
2
Add up the 119909coordinates and halve it
Add up the 119910coordinates and halve it
Find the coordinate of the midpoint joining the points (611) and (15 minus9)
119909 =6 + 15
2
119909 = 105
119910 =11 + minus9
2
119910 = 1105 1
The distance between two
points will always be the hypotenuse
Parallel lines are lines that run equidistant to each other and never
intersect (cross)
Parallel lines have the same gradientDifferent 119910 minus intercepts
119910 = 119898119909 + 119888Same Different
Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)
Substitute in point and solve (119888)
119910 = 2119909 + 119888 2 = 2(4) + 119888
2 = 8 + 119888 minus6 = 119888
119910 = 2119909 minus 6
minus8
Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles
Gradients that are negative reciprocals of each other
Use gradients and coordinate points to calculate the equation of the line
Invert = Reciprocal
1198981 times1198982 = minus1
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 50
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Graphs ndash Contextual graphs
Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to
move away from and return to a point119860
119861
119862
Moving away
Stationary
Returning
Gradient = Speed
119878119901119890119890119889 =119863119894119904119905119886119899119888119890
119879119894119898119890
Not all objects travel at a
constant speed
119866119903119886119889119894119890119899119905 =119903119894119904119890
119903119906119899
Calculate speed at a specific point
by creating a tangent
Velocity - Time graphs record the velocity of a particle moving along a
straight line
The gradient of the line is the acceleration
Area under graph = Distance travelled
Currency Conversions
Cost Comparisons
Predict future costs
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 51
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Graphs ndash Quadratic and Cubic graphs
Quadratic graphs Cubic graphs
-25
-20
-15
-10
-5
0
5
10
15
20
25
-5 -4 -3 -2 -1 0 1 2 3 4 5
Positive quadratic
lsquoursquo shape
Negative quadratic
lsquonrsquo shape
1199092
minus1199092
Substitute the 119909 values into equation to get 119910
and plot points like coordinates
To plot graphs
The points where the graph crosses
the 119909 axis are known as roots
These are the solutions to the
quadratic when it equals zero
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-2 -1 0 1 2 3 4 5 6
Maximum and minimum points
Read off coordinate point from graph
The line of symmetry runs through these points
Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate
Complete the square in the form (119909 minus 119901)2+119902
(119901 119902)Maxmin point can be found by
Can be defined as an equation where the highest power of the
variable (usually 119909) is 3
119910 = 1199093 + 31199092 + 5119909 minus 20
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 52
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Graphs ndash Reciprocal and Exponential graphs
Reciprocal graphs Exponential graphs
119910 =1
119909
119910 =4
119909
119910 =6
119909
As 119886 increases the graphs move
further away from the origin
119909-axis and 119910-axis are asymptotes
Two lines of symmetry
119910 = minus1
119909119910 =
1
119909
119910 =119886
119909
Points can be found by substitution
As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate
When 119896 gt 1 and 119909 is positive
the graph will curve upwards
When 0 lt 119896 lt 1 or 119909 is negative the graph will curve
downwards
119910 =1
119896
119909
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 53
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Graphs ndash Equation of a circle
There is a specific general formula for the equation of a circle
1199092 + 1199102 = 1199032
Algebra Number
1199092 + 1199102 = 25
Centered at Origin
Find the radius (119903)
Radius = 5
1199092 + 1199102 = 1199032
Finding the radius and equation using Pythagoras
52 + 42 = 1199032
41 = 1199032
119903 = 41
1199092 + 1199102 = 41
We may be asked to find the equation of a tangent to a circle at a given point
What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)
Gradient of Radius
Gradient of Tangent
Substitute in Point
119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)
=4
3
Circle TheoremA radius always meets a tangent at a right-angle
minus3
4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)
= Perpendicular line
119910 = minus3
4119909 + 119888
Point 34 4 = minus
3
43 + 119888
25
4= 119888
119910 = minus3
4119909 +
25
4
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 54
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Geometry ndash Perimeter and Area
Perimeter Rectangular areas Triangular areas
The total distance AROUND a 2D shape
Adding all the side lengths together
100119898
100119898
35119898 35119898100 + 100 + 35 + 35
270119898
The process does not change if we have algebraic terms
= 9119909 + 2119910
The total space taken up by a 2D shape
Multiplying two side lengths together
= 119897 times 119908Area of rectangle
6119888119898
12119888119898 Area = 6119888119898 times 12119888119898
721198881198982
With compound shapes break it down
119860 119861 119862
119860 = 601198881198982
119861 = 201198881198982
119862 = 601198881198982
1401198881198982
The area of a triangle takes up half the space of the rectangle that is
formed around it
=1
2(119887 times ℎ)Area of triangle
7119898
4119898
119860 =1
2(7119898 times 4119898)=
1
2(281198982)
141198982
Be sure to use perpendicular heights
Calculate base times height first
Remember to halve your answer
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 55
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Geometry ndash Advanced areas
Parallelogram
Trapezium
Imagine a tilted rectangle
Be sure to use perpendicular heights
= 119887 times ℎℎ
119887
(119886 + 119887)times ℎ=1
2
A more complex formula to know
Add the parallel sides
Halve it
Multiply by height
Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862
Two sides and the included angle
Label triangle
Substitute values in and calculate
119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862
119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)
119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)
119860119903119890119886 = 33471198881198982
We have two sides and the included angle
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 56
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Geometry ndash Circle Definitions
119863119894119886119898119890119905119890119903
119878119890119888119905119900119903
Circumference
Diameter
Radius
Sector
Arc
Tangent
Chord
Segment
The perimeter around the circle
The distance across the centre of the circle
The distance from the centre to the edge of the circle
Part of the area of a circle enclosed by two radii
Part of the circumference of a circle
A straight line that touches the curve of the circle at a point
A straight line segment between two points on the circle edge
The area created by the chord
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 57
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119860 = 1205871199032
Geometry ndash Area and Circumference
Area
Circumference
Sector
Arc length
Pi times the radius squared
Diameter is double the radius
119860 = 120587 times 652
119860 = 120587 times 4225
119860 = 132731198982
119862 = 120587119889
119862 = 2120587119903
119900
119862 = 120587 times 12119888119898
119862 = 377119888119898
The circumference is always about three time the length of the diameter
119860 =119899deg
3601205871199032
Calculate the proportion of the circle required then use area
formula
85deg
85deg
360deg(120587 times 62)
2671198881198982
119871 =119899deg
360120587119889
Calculate the proportion of the circle required then use circumference formula
85deg
85deg
360deg(120587 times 12)
890119888119898
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 58
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Geometry ndash Volume
Prisms
The same cross sectional area throughout
119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ
119860119903119890119886 119900119891 119891119886119888119890 =1
2(8 times 6)
241198881198982
241198881198982 times 10119888119898 = 2401198881198983
Pyramids
The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it
119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905
3
Calculate the volume of the square based pyramid if its height measures 15119888119898
121198881198982 times 15119888119898 = 21601198881198983
21601198881198983 divide 37201198881198983
Spheres
Curved area of cone
119860119903119890119886 = 120587119903119897119897
119903
119881119900119897119906119898119890 =4
31205871199033
Find the volume of the sphere with diameter 16119888119898 Give your answer to
3 significant figures
119881119900119897119906119898119890 =4
3times 120587 times (8)3
21446605851198881198983
21401198881198983
For a hemisphere donrsquot forget to halve your answer
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 59
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Number ndash Approximation and Error Intervals (Bounds)
Error IntervalsBy definition a rounded number does not give us
the exact value
Approximation
Estimates tell us the rough value of a calculation
Rounding off makes it easier to calculate
Round values to 1sf
Perform Calculation
841 times 32
000216asymp8 times 3
0002
24
0002
24000
2
= 12000
Lower Bound
Upper Bound
The minimum a value might be
The maximum a value might be
Continuous Values (Decimal values)
Halve accuracy level
Add on for Upper bound
Subtract for Lower bound
240119898 119905119900 119899119890119886119903119890119904119905 10119898
240119898235119898 245119898
235119898 le 119909 lt 245119898
Discrete values (Whole values)The number of people on a train is 400 to the nearest 100
400350 449
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 60
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Geometry ndash Similarity and Congruence
Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will
need to be calculatedCongruent shapes are just exact
replicas of the original
The angles and side lengths remain the same
The shapes may well be orientated differently
The two shapes are congruentThey are reflections of each other
Similar shapes are just enlargements of the original
The angles remain the same but the lengths have been scaled up or down
This scale factor needs to be calculated in order to solve problems involving similar
shapes
Find two comparative lengths
119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898
3119888119898= 4
Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032
Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033
Work out all scale factors first
Scale factor = 2
Area scale factor = 2 2 = 4
Volume scale factor = 2 3 = 8
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 61
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Geometry ndash Transformations
Reflection Rotation Translation EnlargementReflection the
replacement of each point on one side of a line by the point symmetrically placed
on the other side of the line
Need a mirror line
Reflection in the line119909 = minus2
Rotation the action of rotating a shape about an
axis or central point
Need a measure of turn
Need a Centre point
Need a Direction
Use of tracing paper
Rotation90deg
Clockwise
About (minus50)
Translation the action of lsquoslidingrsquo a shape to a new
position
Described using Vector Notation
119929 rarr 119930 =
Translation minus108
Enlargement the action of resizing a shape to the scale factor given from a specific
point
Need a Centre point
Need a scale factor119873119890119908 119904ℎ119886119901119890
119874119903119894119892119894119899119886119897 119904ℎ119886119901119890
EnlargementScale factor
3
From (3 minus6)
119929 rarr 119930
Use projection
lines
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 62
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Geometry ndash Congruence criteria for triangles
Side Side Side Side Angle Side Angle Side Angle Right Hypot Side
All three sides of one triangle are equal to the
corresponding sides of the other triangle
Two sides and the included angle are equal
to the corresponding sides and included angle of the
other triangle
Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle
Each triangle contains a right angle the hypotenuses
are equal in length as well as another equal comparative side
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 63
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Geometry ndash Constructing bisectors and Loci
Create arcs above and below the line
Set compass to over halfway
Sometimes you will need to
create a separate line segment
Mark off each line segment
Create arcs within the angle
Do not alter the compass at all
Do not alter the compass at all
The locus of points from a point is a circle
119860 119861
The locus of points between two points is the perpendicular
bisector
The locus of points between two lines is
the angle bisector
Notice how the locus of points around a corner is CURVED
A locus is a series of
points that satisfy a
particular conditionLoci is the
plural and will often involve
several conditions
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 64
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Algebra ndash Solving by factorising and the Quadratic formula
Solving by factorising The quadratic formula
Substitute values into the formula to generate two answers for 119909
119909 =minus119887 plusmn 1198872 minus 4119886119888
2119886
51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888
The formula you need to
know
119909 =minus 8 plusmn 8 2 minus 4 5 minus4
2 5
119909 =minus8 plusmn 144
10
119909 = 04 or minus2
Substitute and simplify
Carry out two calculations
1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0
Factorise the quadratic ndash You may need to rearrange first
1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0
Find values for 119909 that will make each bracket = 0
119909 = minus7 119900119903 minus 1
21199092 minus 2119909 = 3 1 minus 119909
21199092 + 119909 minus 3 = 0
21199092 minus 2119909 = 3 minus 3119909
2119909 + 3 119909 minus 1 = 0
119909 = minus3
2119900119903 + 1
Expand and rearrange to = 0
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 65
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Algebra ndash Completing the square and solving quadratics
Changing the form of the quadratic
1198861199092 + 119887119909 + 119888
Completing the square 119886 ne 1
(119909 + 119901)2+1199021199092 + 119887119909 + 119888
119909 + ൗ119887 2
2minus ൗ119887 2
2+ 119888
119909 + 119901 2 + 119902
Halve the
Simplify
1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2
119909 + 3 2 minus 11
1199092 minus 10119909 + 15 = 0
coefficient of 119887
Solving equations by completing the square
119909 minus 5 2 minus 10 = 0
119909 minus 5 2 = +10 119909 minus 5 = plusmn 10
119909 = 5 plusmn 10119909 = 5 + 10 = 816
119909 = 5 minus 10 = 184
Complete the square Solve equation
31199092 + 6119909 minus 4 = 0Factor out the three
3 1199092 + 2119909 minus4
3= 0
Complete the square on this expression
3 119909 + 1 2 minus 1 2 minus4
3= 0
3 119909 + 1 2 minus7
3= 0
3 119909 + 1 2 minus 7 = 0
Complete the square
Expand
Solve for 119909
3 119909 + 1 2 minus 7 = 0
3 119909 + 1 2 = 7
119909 + 1 2 =7
3
119909 + 1 = plusmn7
3 119909 = minus1 plusmn7
3119909 = 0528 119900119903 minus 2528
Add 7 to both sides
Divide both sides by 3
Square root both sides
Subtract 1from both
sides
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 66
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Algebra ndash Simultaneous equations
Simultaneous equations
Linear equations (Elimination method)
Linear equations (Substitution method)
Quadratic equations (Substitution method)
Multiply equations to get matching coefficients
4119909 + 3119910 = 53119909 + 2119910 = 4
times 3
times 4
12119909 + 9119910 = 1512119909 + 8119910 = 16minus
119910 = minus1
Addsubtract equations
Substitute to find second variable
3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2
3119909 minus 2 = 4 119909 = 2
Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)
119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7
119910 = 2119909 + 1Rearrange to get a single variable on
its own
3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1
16119909 + 3 = 7119909 = 025
Substitute equations to find
first variable
Substitute 119909 = 025 into equation 2
119910 = 2(025) + 1 119910 = 15Substitute to find second variable
119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2
119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4
119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4
119910 = (minus1) + 6119910 = 5
119910 = (4) + 6119910 = 10
Substitute equation into quadratic and rearrange to = 0
Factorise and find two solutions for variable
Substitute each answer to find other pair of solutions
Equations involving two or more unknowns that are to have the same values in each equation
4119909 + 3119910 = 53119909 + 2119910 = 4
2119909 minus 3119910 = 45119909 + 2119910 = 1
3119910 + 10119909 = 7119910 = 2119909 + 1
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 67
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Algebra ndash InequalitiesSymbols
Inequalities on a numberline
Solving Quadratic inequalities
Shows the range of values for an inequality
Open circle does not include value
Closed circle includes value
Solving Linear inequalities
Same process as solving linear equations
When we multiply or divide by a negative number we must flip the inequality sign
119909 + 5 le 16
119909 le 16 minus 5
119909 le 11
minus2119909 gt 4
119909 lt minus2
4 lt 3119909 + 1 lt 16
1 lt 119909 lt 5
Same process as solving equations
Sketch the graph to interpret the range of values required
1199092 le 16
119909 = plusmn4The roots
minus4 le 119909 le 4The range of values
below the line
Two variable inequalitiesInequalities with 2-variables need to be
represented on a graph
Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3
le - Solid linelt - Dashed line
Draw the line graph for each inequality
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 68
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Statistics ndash Probability
Introduction
Counting outcomes
Calculating probability Types of events
The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904
119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904
Simplify answers where
possible
Mutually exclusive
Working out how many combinations there are
1 2 3 4 5 6
Heads H 1 H 2 H 3 H 4 H 5 H 6
Tails T 1 T 2 T 3 T 4 T 5 T 6
Rolling a die and flipping a coin
This is a sample space diagram
There are 12 possible outcomes from this event
119875(3) =2
8
1
4
The lsquoORrsquo rule (mutually exclusive)
119875 119886 119900119903 119887 = 119875 119886 + 119875 119887
119875 2 119900119903 4 =2
8+1
8
3
8Add each
probability
The lsquoANDrsquo rule (independent)
119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887
119875 2 119905119886119894119897119904 =1
2times1
2
Flip a coin twice1
4
Multiplyeach
probability
Events that cannot happen at the same time
Rolling a die 119875(1 119886119899119889 6)
All probabilities from the event will sum to make 120783
Independent events
Events where the outcome of one doesnrsquot affect the outcomes of the others
Picking a counter out of a bag replacing it and repeating
Dependent events
Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not
replacing it and repeating
Calculating expected outcomes
119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 69
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Statistics ndash Venn Diagrams and Probability trees
Venn diagrams
Intersections cap and Unions cup
A set is a collection of things called elements
119860 = 2 3 5 7 11The set of prime
numbers less than 12
119860 = 2 3 5 7 11 119861 = 1 3 5 7 9
119860 cap 119861 = 3 5 7 minus Intersection of A and B
119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B
119860prime cap 119861 = 1 9
The crossover between sets
All values in the sets
Values in 119861 but not in 119860
Probability trees
Dependent events
119875 119866 =85
200
119875 119867 cap 119866 =25
200
119875 119867 cap 119866prime =50
200
Probability trees are really useful to calculate the probabilities of combined events happening
Multiply along branches
119875 119877119890119889 119886119899119889 119877119890119889 =4
15
119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2
15+
6
15=
8
15
Add together all combinations
Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc
119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012
When dealing with no replacement remember to reduce the denominator
by one for the second event
119875 119874119899 119905119894119898119890 = 018 + 056 = 074
copy EzyEducation ltd 2017
103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
copy EzyEducation ltd 2017
Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
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Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 70
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103104
Number ndash Standard Form
AddSubtract Standard formTake numbers out of Standard form
AddSubtract valuesConvert answer back to Standard form
Basic Structure
119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903
283 times 106 = 2830000
314 times 10minus4 = 0000314Positive power of 10 = Large number
Negative power of 10 = Small decimal number
(323 times 104) + (82 times 103)
32300 8200+
40500
405 times 104
MultiplyDivide Standard formSeparate the numbers and powers of 10
MultiplyDivide numbers Apply laws of indices to power of 10s
Give answer in Standard form
( times ) times ( times )46 3
46 times 3
138
103104 times
107times
138 times 108
156 10minus4 75 10minus7( times ) divide ( times )
156 75divide 10minus4 10minus7divide
0208 103
208 102
times
times
===
times
times
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
copy EzyEducation ltd 2017
Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
copy EzyEducation ltd 2017
Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
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Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
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Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 71
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Algebra ndash Algebraic fractions
Multiplication and DivisionAddition and Subtraction
Make sure the denominators are
the same
119909
119910+3
119910119909 + 3
119910=
1199092 + 5
1199102+119909
119910
Multiply fractions to achieve same denominator
times 119910
times 119910
1199092 + 5
1199102+119909119910
11991021199092 + 5 + 119909119910
1199102
1199092
9 119909 minus 5minus119909 + 4
119909 minus 5
Same rules apply in Subtraction
times 9
times 9
1199092
9 119909 minus 5minus9 119909 + 4
9 119909 minus 5
1199092 minus 9119909 minus 36
9 119909 minus 5
Expand brackets and simplify where possible on the numerators
2
119909 + 1minus
5119909
119909 minus 4
2 119909 minus 4
119909 minus 4 119909 + 1minus
5119909 119909 + 1
119909 minus 4 119909 + 1
minus8 minus 51199092 minus 3119909
119909 minus 4 119909 + 1
2 119909 minus 4 minus 5119909 119909 + 1
119909 minus 4 119909 + 1
Multiply across the numerators and denominators
Cross cancel terms where possible
Simplify each fraction
Factorise expressions
2
119909times1199092
119910
2
119909times1199092
119910
119909
1
2119909
119910
6119886 + 6119887
2times
1
119886 + 1198876 119886 + 119887
2times
1
119886 + 119887
119865119886119888119905119900119903119894119904119890= 3
To divide multiply by reciprocal of 2nd fraction
4119910119911
119909divide1199101199112
10
119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911
119909times
10
1199101199112
4119910119911
119909times
10
1199101199112119911
40
119909119911
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Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
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RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
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Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
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Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
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Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
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Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
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Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 72
copy EzyEducation ltd 2017
Algebra ndash FunctionsThink of it as a machine that has an input which is
processed by the function to give an output
119891 4 = 6 119891 minus3 = minus1
Substitute input into the function to
generate output value
119892 119909 = 4119909 minus 3 find 119892 21
Using functions
119892 21 = 4119909 minus 3
119892 21 = 81 119892 21 = plusmn9
Substitute input into the function and calculate
119891 119905 = 31199052 + 2 find 119891 2
119891 2 = 31199052 + 2Substitute input into the
function and calculate
119891 2 = 12 + 2 119891 2 = 14
Inverse functionsA function that performs the
opposite process of the original function
You have been given the output and need to work out
the value of the input
119891 119909 119891minus1 119909Normal function Inverse function
Two ways to solve problems involving inverse functions
Function machine method Subject of Formula method
Composite functions
119891 119909 = 5119909 + 2
119910 = 5119909 + 2 119910 minus 2 = 5119909
119910 minus 2
5= 119909119891minus1(119909) =
119909 minus 2
5
The combination of two or more functions to create a new function
119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)
The output of 119892(119909) will form the input of 119891(119909)
119892 119909 = 119909 minus 2
119891 119909 = 2119909 + 2
119891 119909 minus 2 = 2119909 + 2
119891 119909 minus 2 = 2 119909 minus 2 + 2
119891119892 119909 = 2119909 minus 2
copy EzyEducation ltd 2017
RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
copy EzyEducation ltd 2017
Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
copy EzyEducation ltd 2017
Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
copy EzyEducation ltd 2017
Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
copy EzyEducation ltd 2017
Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 73
copy EzyEducation ltd 2017
RPR ndash Rates of change
Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity
It is represented by the Gradient of a line
Gradient =1199102minus1199101
1199092minus1199091
Gradient =119877119894119904119890
119877119906119899
Interpreting Rates of Change
GradientAmount of (119910) per
Amount of (119909)
Rate of change = $50 per month
Instantaneous rate of change
The rate of change over a given interval
The rate of change at a particular moment
Create chord between two intervals
Calculate gradient of chord
Interpret gradient as a rate of change
Create tangent at specific point
Calculate gradient of tangent
Interpret gradient as a rate of change
copy EzyEducation ltd 2017
Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
copy EzyEducation ltd 2017
Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
copy EzyEducation ltd 2017
Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
copy EzyEducation ltd 2017
Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 74
copy EzyEducation ltd 2017
Graphs ndash Translations and Reflections
ReflectionTranslation
A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position
119891 119909 + 119886The graph
shifts updown the 119910 minus 119886119909119894119904 by 119886 units
119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904
by 119886 units
+= Left
minus= Right
Adding to the function causes a Translation
Given 119891(119909) Sketch 119891(119909 minus 3) + 5
3 units right 5 units up
119891(119909)
119891(119909 minus 3) + 5
3
5
Reflection The replacement of each point on one side of a line by the point symmetrically placed on
the other side of the line
The graph is reflected in the 119962 minus 119938119961119946119956
119910 = minus119891(119909) 119910 = 119891(minus119909)
The graph is reflected in the 119961 minus 119938119961119946119956
The outputs are reversed
The inputs are reversed
119891(119909)119891(minus119909)119891(119909)
minus119891(119909)
copy EzyEducation ltd 2017
Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
copy EzyEducation ltd 2017
Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
copy EzyEducation ltd 2017
Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 75
copy EzyEducation ltd 2017
Graphs ndash Using graphs to find solutions
When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically
2119909 + 3119910 = 18
119910 = 119909 + 1
Solve simultaneously or graph each
equation
The point of intersection will be
your solutions The point where 119909 and 119910 have the same value
for each equation
119909 = 3 119910 = 4
119910 = 1199092 minus 2
119910 = 2119909 + 1
Solve simultaneously or graph each
equation
119909 = 3 119910 = 7
119909 = minus1 119910 = minus1
Two intersections = two sets of solutions
or
Find the solutions of 119909 when
1199092 minus 2 = 7
119909 = 3
119909 = minus3
Plot graph and read off points
or
Identify and read off the points of intersections for your solutions
Use graph to read off specific values
for 119909 and 119910
Plot each equation
separately
copy EzyEducation ltd 2017
Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
copy EzyEducation ltd 2017
Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 76
copy EzyEducation ltd 2017
Graphs ndash Estimating gradients Area under a curve
Area under a curveEstimating GradientsCalculate average gradient from
beginning to end These are not very accurate and do not
show the full picture
Break the graph down into smaller pieces to see what is happening
119866119903119886119889119894119890119899119905 119860 = ൗ1 3
119866119903119886119889119894119890119899119905 119861 = ൗ5 3
119866119903119886119889119894119890119899119905 119862 = ൗ3 5
03119898119904
17 119898119904
06119898119904
Find out what is happening at a particular point - Tangents
Estimate because tangents vary
The area under a curve will enable you to estimate the total distance travelled in
velocity time graphs
1
2(119886 + 119887) times ℎ
Trapezium Triangle1
2(119887 times ℎ)
Formulae needed
Estimate the distance travelled for the first ten seconds
275119898
copy EzyEducation ltd 2017
Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 77
copy EzyEducation ltd 2017
Geometry ndash Sine and Cosine rules
The Sine rule formula The Cosine rule formula
119886
sin119860=
119887
sin119861=
119888
sin119862
sin119860
119886=sin119861
119887=sin119862
119888or
Missing length Missing angle
Using Trigonometry to calculate missing angles and side lengths in non right angled triangles
We tend to use the Sine rule if we know an angle and its opposite length
35
sin(125)=
119909
sin 35 times sin(35)
35
sin(125)times sin 35 = 119909 2451119898
sin(119909deg)
28=sin(40)
21times 28
sin 119909 =sin(40) times 28
21119909deg = sinminus1
sin 40 times 28
21
119909 = 59deg
1198862 = 1198872 + 1198882 minus 2119887119888 cos119860
cos119860 =1198872 + 1198882 minus 1198862
2119887119888
Missing length
Missing angle
If we know two sides AND the included angle
1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))
1199092 = 6964978hellip
119909 = 8346119896119898
cos 119909 =60 2 + 35 2 minus 90 2
2 60 90
cos 119909 = minus0303hellip
119909 = cosminus1(minus0303hellip) 119909 = 1077deg
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 78
copy EzyEducation ltd 2017
Graphs ndash Trigonometric Graphs
Sine 119909 axis intercept at 0deg 180deg 360deg
Cosine
Tangent
119878119894119899 120579 =119900119901119901
ℎ119910119901Turning points at 90deg 270deg
They are periodicalthey will continue to repeat every 360deg in both directions
There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)
Use symmetry of graphs to solve problems
119862119900119904 120579 =119886119889119895
ℎ119910119901
119909 axis intercept at 90deg 270deg
Turning points at 0deg 180deg 360deg
119879119886119899 120579 =119900119901119901
119886119889119895
119909 axis intercept at 90deg 270deg
Asymptotes at 90deg 270deg
copy EzyEducation ltd 2017
Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 79
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Statistics ndash Types of data and Sampling
Types of dataQuantitative Qualitative
Data that is numeric Data that is descriptive
Discrete
Continuous
Data that can be counted and only has
certain valuesPeople on a bus
Shoe SizeDress size
Data that can be measured to various
levels of accuracyHeight of a treeSpeed of a car
Mass of a person
Categorical data
Data which can be grouped into
categories
Grouped Data
Hair colourFavourite food
Sport
Data which is organised into classes
Primary
Secondary
Data collected by you
Data gathered from another source
SamplingWe collect and analyse data to give us information about a population
Census
Sample
Bias
Data is collected from the WHOLE population
Can take a very long time to collect the information
Data is collected from PART of the population
Quicker to collect the data and the data can be used to describe the whole population
Random Stratified
Each member assigned a numberNumbers randomly generatedThose numbers used in sample
119860119898119900119906119899119905 119894119899 119892119903119900119906119901
119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890
Your sample is randomly selected Proportionate numbers from each group selected to make sample
Some situations can cause bias and make the sample
unrepresentative
When and where the sample is takenIs the sample large enough
Who is in the sample
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
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Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 80
copy EzyEducation ltd 2017
Probability and Statistics ndash Frequency and Two-Way Tables
Colour Tally Frequency
Yellow 1
Blue 7
Red 7
Green 3
Pink 2
Sum 20
Yellow Blue Red Red
Red Blue Green Pink
Blue Red Blue Green
Pink Red Blue Red
Red Blue Green Blue
Favourite ColourTally Up Add Tally Marks
Relative Frequency
120 = 005
720 = 035
720 = 035
320 = 015
220 = 01
Relative frequency describes
what proportion
selected that colour
If the sample size is large enough we
can interpret relative
frequencies as
probabilities
TwondashWay Frequency Tables
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Provides information about the frequency of two variables
Number of girls in Year 12 is 70
Year 12 Year 13 Total
Male 120 80 200
Female 70 100 170
Total 190 180 370
Rows and columns add up
Use to calculate missing values
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 81
copy EzyEducation ltd 2017
Geometry ndash Circle Theorems
Angle at the centre is twice the angle at the circumference
119909
2119909
Look for the lsquoArrowrsquo Shape
Angles in the same segment are equal
119909
Look for the lsquoBowrsquo Shape
119909
119909
Angle subtended at circumference by a semicircle is 90deg
Opposite angle to the diameter
90deg
Opposite angles in a cyclic quadrilateral sum to 180deg
A
C
B
D
A + C = 180deg B + D = 180deg
Tangents and radii meet at 90deg
90deg
Tangents from a point have equal length
Alternate Segment Theorem
119909
119909119910
119910
Tangent just touches the
circumference
Tangent
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 82
copy EzyEducation ltd 2017
Number ndash Surds and Rationalising the denominator
Surds Rationalising the denominatorSurds are expressions which contain an irrational square
root Rationalising the denominator involves removing all
of the roots from the bottom of a fraction
119886 times 119887 = 119886 times 119887
119886
119887=
119886
119887
3 times 7 = 3 times 7 = 21
6
10=
6
10
3
5 =3
5
Writing in the form 119886 119887
Think square numbers 200
119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100
Choose the largest square factor
100 times 2 = 10 2
119886 + 119887 ne 119886 + 119887 5 + 20 = 25
6
3
Multiply top and bottom by irrational root
6
3times
3
3
6 3
3
6 3
9
5
3 + 2times3 minus 2
3 minus 2
5
3 + 2
Multiply top and bottom by Conjugate
(opposite root)
=5 3 minus 2
3 + 2 3 minus 2Expand and simplify
=15 minus 5 2
9 minus 3 2 + 3 2 minus 2=
15 minus 5 2
7
A more complex denominator
copy EzyEducation ltd 2017
Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
copy EzyEducation ltd 2017
Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 83
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Geometry ndash Vectors
Vectors describe translations
119909119910
Direction along
Direction updown
Represented by arrows
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
24
119938 =119938
3minus2
119939 =
119939
2 right
4 u
p
3 right
2 d
ow
n
Notation
119860119861 119938 119938Magnitude =
Length of Arrow
Direction = Where arrow is
pointing
Adding and Subtracting
Start arrow at end of previous vector
119938 119939
119938 + 119939119939 starts at the
end of 119938
Negative vector goes in opposite direction
119939
minus119939 119938 119939
minus119939
Multiplying
Only affects magnitude not direction
3119939 = 119939 + 119939 + 119939
119939
1199393119939
Parallel VectorsSame direction May have different
magnitude
119938 minus 3119939 3(119938 minus 3119939)
119939
Same Direction
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 84
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Number ndash Units ndash Mass Length Area and Volume
Length
Mass Area and Volume
119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000
divide 1000 divide 1000 divide 1000
11198961198921000119892 =
1000119898119892 = 1119892
1000119896119892 = 1 119905119900119899119899119890
Standard Metric
Conversions
119898119898 119888119898 119898 119896119898times 10 times 100 times 1000
divide 10 divide 100 divide 1000
1000119898 = 1 119896119898100119888119898 = 1119898
10119898119898 = 1 119888119898Standard Metric
Conversions
Area is a 2D measurement formed by multiplying two lengths
100001198881198982 = 111989821001198981198982 = 11198881198982
10000001198982 = 11198961198982
Volume is a 3D measurement formed by multiplying three lengths
10000001198881198983 = 1119898310001198981198983 = 11198881198983
10000000001198983 = 11198961198983
Squared units results in squared conversion factors
Cubed units have cubed conversion factors
copy EzyEducation ltd 2017
Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 85
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Number ndash Units ndash Time and Money
MoneyTime
1 Week 7 days
1 Day 24 hours
1 Hour 60 minutes
1 Minute 60 seconds
Conversions between units are not decimal
Convert 350 seconds into minutes and seconds
1 Minute 60 seconds
2 Minutes 120 seconds
3 Minutes 180 seconds
4 Minutes 240 seconds
5 Minutes 300 seconds
6 Minutes 360 seconds
5 minutes and 50 seconds
Add 6 Τ1 4 hours to 4 23 119886119898
4 23 119886119898+6 ℎ119900119906119903119904
10 23 119886119898+15 119898119894119899119904
10 38 119886119898
Adding and Subtracting money requires lining up the decimal point
pound 6 5 2 0pound 1 0 0 0pound 0 6 5+
5857pound
pound1 is worth and can be exchanged for $134
119866119861119875 119880119878119863times 134
divide 134
Money comes in specific denominations
Adding and Subtracting money requires lining up the decimal point
GBPUSD = 1 134
Exchange rate calculations
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg
Page 86
copy EzyEducation ltd 2017
Geometry ndash Bearings
Angle clockwise from North
119873
75deg
075degClockwise
119873
310deg
310deg
Clockwise
Always 3 digits
75deg 075deg
4deg 004deg
Sentence Structure Important
The bearing ofB from A is 075deg
119873
75deg
B
A
Lines North are Parallel
119873
B
A
119873
75deg
Co-Interior Angles
119873
B
A
119873
75deg
105deg
120579
75deg + 120579deg = 180deg
120579deg = 180deg minus 75deg = 105deg
Angles around a point
119873 B
A
119873
75deg
105deg
120601
105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg
255deg
The bearing of A from B is 255deg