Mathematics sec 1

MATHEMATICS secondary 1

Nicco Alyssha Parikh

PRIME NUMBERS Prime numbers : Numbers that can only be divided by 1 and itself . Composite numbers : Not prime numbers . Prime numbers from 1-20 = 2,3,5,7,11,13,17,19 * 1 is not a prime number – because it can only be divided by itself . Q.) How many prime numbers are there from 1-100? A.) 25.

Step by step finding Prime numbers . - Cross out the number 1 - Circle the number 2 and cross out all the other multiples of 1 . - Circle the number 3 and cross out all the other multiples of 3 . - Circle the number 5 and cross out all the other multiples of 5 . - Circle the number 7 and cross out all the other multiples of 7 . - Continue the process unit all unit all the numbers are either circled or

crossed.

HIGHEST COMMON FACTOR(common)How to find the highest common factor?

Find the highest common multiple of 15 and 75 ?

15,75 5,25 1,5

HCF = 3x5=15

35

Express 252 in PRIME FACTORS

252

2 x 126

2 x

2 X 63

22 3 X 21

3 3 x 7

From the above factor tree, We have 252 = 2x2x3x3x7

INDEX NOTATION Index notation is using the power of a certain number.e.g.) 252= (first, prime factorise the numbers)= 2x2x3x3x7= 22 x 32 x 7

12= 2x2x3 can be written as 12= 2 x 32

LOWEST COMMON MULTIPLE(max) Lowest common multiple of 65 , 175 , 135

65,175,135

Please note that we have to arrive to the answers to all be one at the last ladder

13, 35,271,35,271,1,27

1,1,1

5133527

LCM(LOWEST COMMON MULTIPLE)= 5x13x35x27 =61425

FINDING CUBE ROOT AND SQUARE ROOTS

1) FIRST PRIME FACTORISE THE NUMBER

EG.) square root of 144 = 24 x 32

2) Arrange them into 2 brackets

Square root ( 2x2x3) (2x2x3)

3) Solve what is in 1 bracket

2x2x3=4x3=12

Cube root

Do the same only at step to , instead of 2 brackets , it becomes 3 brackets .

IntegersPositive and Negative integers. In the number line , the more left you go , the larger

the number gets(smaller value) .. Zero is an integer by itself- not positive or negative.*note that there is no such thing as +0 or -0 .- BODMAS rule stated that everything should be from

left to right UNLESS there is a bracket .

ADDITION OF INTERGERS3+2=5 3+(-3)=0-3+4=-1-4+(-2)=-6

Owe someone 4 dollars and another 2 dollars.

SUBTRACTION OF INTEGERS-7+(-11)+9=-7-11+9=-18+9= -9

34+(-18)+9=34-18+9=16+9=25

MULTIPLICATION OF INTEGERS+x+=+-x+=--x-=+-x0=? (0)

-2x3x(-1)=-6x(-1)=6

DIVISION OF INTEGERS

3x6 3-2= (18 3)-2=6-2=4

6 2x4 + (-3)= 3 x 4 +(-3)= 12 + (-3)= 9

*ALWAYS DO FROM LEFT TO RIGHT

ALWAYS DO THE “ POWERS “ FIRST

(-4)2 (-8) + 3 x (-2)3

= 16/(-8) + 3 x (-8)= -2 + 3 x (-8 )= -2 + (-24)=-26

RATIONAL NUMBERS

a/b - b cannot be “0”

e.g : mixed numbers improper fraction

- Using the cancellation method ….Such as: - 21/17 X 19 /7 = - ?

THE INTEGERS IN RATIONAL NUMBERS CAN BE BOTH POSITIVE AND NEGATIVE.THE CHANGING OF SIGNS MUST BE INCLUDED!!!!! REMEMBER WHEN DIVIDING A FRACTION OR FRACTIONS , SAME-CHANGE-INVERT !! ALSO REMEMBER THAT EVEN IF THERE ARE 3 OR MORE FRACTIONS ONLY ONE DOESN’T CHANGE –

DURING DIVISION OF FRACTIONS ONLY !! DENOMINATORS MUST BE THE SAME.

**Irrational CANNOT BE EXPRESSED AS A FRACTION

ALGEBRA

• Actually writing numbers in the form of letters• IF YOU ARE 40 YEARS OLD , I AM 20 YEARS

YOUNGER THAN YOU , MY AGE WILL BE (40-20) .

• BUT IF I AM x YEARS OLD , YOU ARE (x-20)years old

• OF BOTH , POSITIVE AND NEGATIVE INTEGERS . THE (-)MINUS SIGN IS ACTUALLY THE “NEGATIVE” SIGN .

ALGEBRA

Only like terms can combine into a single term ( BY ADDITION OR SUBTRACTION ONLY )

Like terms : 1) ab , 2 ab ( yes)2) x , 2x2 (no)3) 3p,7p (yes)4) xy , 2x2y (no)

SUBTRACTION IN ALGEBRA

1) (+)3a-2b+2a-3b= 3a+2a – 2b – 3b

= 5a-5b

2)[3a+3b(a-bc)] FROM 3a-3b=(2a-3b) – (3a2 -3b2c)=2a-3a2-3b-3b2c=-1a3-3b3c

Step 1 : rearrange

Step 2 : evaluate

DIVISION IN ALGEBRA

27a / 3a= 9 ( cancel the “a”)

27/3a=27/3a

DIVIDED AWAY

TERMS , VARIABLE , COEFFICIENTWhen x = 4 , When y = 6 When z = 10 ,

x+y= 4+6 = 10 z-(x+y) = 10-10 = 0 5x = 5x4 = 20

THE VALUE OF x IS CALLED A VARIABLE 5 IS ATTACHED TO x , SO 5 is the coefficient OF x.

E.G) 10a _ a is a variable and 10 is the coefficient OF a 6ab - ab is a variable and 6 is the coefficient of ab 2B - 2 of B’s B2– 1b 2so , B is the variable and 1 is the coefficient of b2

ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS

RECALL : addition / subtraction of integers

e.g.) sum of 4 and 2 = 4+2 = 6 Subtract 2 from 5 = 5-2 = 3

Exponents often are used in the formula for area and volume. In fact, the wordsquared comes from the formula for the area of a square.

s

s

Area of a square: A = s2

The word cubed comes from the formula for the volume of a cube.

ss

s

Volume of Cube: V = s3

SQUARE ROOTS AND CUBE ROOTS

FACTORISATION

4p2 + 2pq= 2p(2p+q)

Common factor

1) Factorisation is the process of finding a term or an algebraic expression.

2) The common factors of several algebraic terms are numbers or terms that are the factors of all algebraic terms

3) An algebraic expression with 2 or more terms can be factorised by taking out all the common factors of the expressions from the brackets.

2xy + 6y + 3x +9 = 2y(x+3)+3(x+3)

=( 2y+3)(x+3)Same

FACTORISATIONFactorisation means taking out the common factors . Factorisation is NOT expansion . Factorisation vs expansion => opposite

OPERATION OPPOSITE

ADDITION SUBTRACTION

SQUARE SQUARE ROOT

FACTORISTION EXPANSION

CUBE CUBE ROOT

DIVISION MULTIPLICATION

EXPANSIONExpansion – final answer should not have fractions . (Using the “rainbow” method )

e.g) 3(2+x) = 6+x e.g) -3(2h-2k)+4(k-3h) = -6 -6k +4k – 12h = -6h-12h+6k+4k = -18h+10k

STEP 1 : Remove the bracket by doing EXPANSION .

STEP 2 : Rearrange to put the “like” terms together

NOTE : 2 SETS OF BRACKETS , 2 EXPANSIONS

ALGEBRA

Square root is the opposite of squareE.G.) p(square) is opposite of p

-DETAILS MUST BE STATED CLEARLY- Times (x) must be written in “bracket format” such as 3x4=

3(4)

2P= 2 x PP2= P x PP3= p x p x p3P= 3 X P

What algebraic expression can be used to find the perimeter of the triangle below?

a b

c

Perimeter = a + b + c

In this algebraic expression, the letters a, b, and c are called ________.variables

In algebra, variables are symbols used to represent unspecified numbers or values.

NOTE: Any letter may be used as a variable.

Variables and Expressions

It is often necessary to translate verbal expressions into algebraic expressions.

English word(s) Math Translation

more than

less than

product

addition

subtraction

multiplication

of multiplication

quotient divisionWrite an algebraic expression for eachverbal expression:

a) Eight more than a number n. 8 + ntranslates to

b) Seven less the product of 4 and a number x. 4x-7 translates to

c) One third of the size of the area a. translates to or a3

1

3

a

Variables and Expressions

Find the perimeter of the triangle.If a is 8 , b is 15 and c is 17

a b

c

Perimeter = a + b + c Write the expression.

= 8 + 15 + 17 Substitute values.

= 40 Simplify.

= 8

= 17

= 15

SUBSTITUTION

FINDING THE UNKNOWN

e.g.) 3x – 2 = 43x= 4+23x=6x = 6/3X=2

(+)11-2k=17 -2=17-11 -2k=(-2) 17-11=6 6/-2 = -3(k)K=-3

2h +1.3=2.82h=2.8-1.32h=21.5-2=0.5h= 0.5

*If “ –” , do “+” If “x” do “/”

FINDING THE UNKNOWN II

3.14 => recurring number

FURTHER EXAMPLES ON EQUATIONS7 + 2x = 6x-52x=6x-5+72x=6x-122x-6x=-12-4x=-12X= -12/-4= +3

.

6hx + 12ky +9kx +8hy =6hx + 9kx + 12ky + 8hy = 3x (2h+3k) + 4y (3k+2h)=(3x+4y) (2h+3k)

*REARRANGETHE ONE WITH THE MOST COMMON FACTOR

ESTIMATION

1003 x 78 ~ 1000 x 80 = 80,000

1003 x 85~ 1000 x 90 = 90,000

~

~

*LESS THAN 5- ROUND DOWN / ignore (“0”)

*5 OR MORE – ROUND UP

1300 + 6~ 1000+10= 1010~

AREA AND PERIMETER

AREA)Triangle = ½ x base x height Rectangle = Length x Breadth Square = Length x Length Circle= π x radius x radius (πr2)Parallelogram = Base x height (perpendicular height)Trapezium = ½ x (a+b) x height (a&b 2 parallel lines)

AREA AND PERIMETER

PERIMETER)Triangle = plus (+) all outer sidesRectangle = plus(+) all outer sidesSquare = plus (+) all outer sides Circle= (circumference) π x diameter (πD)Parallelogram = Plus(+) all outer sidesTrapezium = plus(+) all outer sides

FORMULAS FOR MEASURING VOLUME

CUBE = Length x Length x Length CUBOID = Length x Breadth x Height PRISM = Base area x Height = 1/2 x Length x Breadth x HeightPARALLELOGRAM = Base x Height

CONE = 1/3 x x radius2 x heightSPHERE= 4/3 X x radius3

NUMBER SEQUENCENUMBER SEQUENCE PATTERN2,4,6,8,10,12 2 times table

1,3,5,7,9,11 Odd numbers/add 2

1,2,4,8,16,32 Power of 2

2,5,8,11,14,17,20 Add 3

0,10,20,30,40,50,60… Add 10 / 10 times table

1,3,6,10,15 Add 1 to the top

1,1,2,3,5,8,13,21 Add the 1st 2 numbers to get the 3rd number

FINDING SEQUENCES

1st layer 1 = 12nd lay+0er 1+2= 33rd layer 1+2+3=64th layer 1+2+3+4= 10

30th layer ?1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30 = 465

FINDING SEQUENCESStep 1 : Find the pattern Step 2 : See how the pattern flows Step 3 : Continue the pattern

SOLVING INEQUALITIES

Symbol Words Example> greater than x + 3 > 2< less than 7x < 28≥ greater than or equal to 5 ≥ x - 1≤ less than or equal to 2y + 1 ≤ 7

SOLVING INEQUALITIES12 < x + 5If we subtract 5 from both sides, we get:12 - 5 < x + 5 - 5 7 < xBut put an "x" on the left hand side ... so let us flip sides (and the inequality sign):x > 7Do you see how the inequality sign still "points at" the smaller value (7) ?

ANS: x > 7

VOLUME Volume of cuboid Length x breadth x heightVolume of cube Length x Length x Length Volume of pyramid 1/3 x Base x Height Volume of Cylinder Base x Height Volume of cone1/3 x Base x Height Volume of sphere 4/3 x π x r3

UNIT CONVERSION

Units : mm,cm,m,km,ha (perimeter) : mm2,cm2,m2, km2,ha2 (area)

10mm= 1cm1mm=0.1cm100cm= 1m1cm=0.01 m1000mm= 1m 1mm= 0.001 m 1 ha = 10000 m2

1kg=1000g (1k-1000 , g – grams)

VOLUME AND TOTAL SURFACE AREA

1. CUBE Volume : length3 Area : 6xlength2

2. CUBOID Volume : length x breadth x height Area: 2(lb + bh + hl ) 3. PRISM Area : Base area x height Volume: (Perimeter of base x h ) + 2base area 4. Cylinder Volume : πr2h Area: 2πr2 + 2πrh

UNIT CONVERSION

185mm= 185 x 0.1 cm = 18.5 cm21cm = 21 x 10mm = 210 mm21cm = 21 x 0.01m = 0.21 cm1 hectare = ?x?

CONVERSION

1m = 100cm (x100)1cm = 0.01m (/100)1m=0.001km(/1000)1000m =1km (x1000)1hour=60mins 1minute=60 seconds

RATIO (REPEATED IDENTITY)If a:b = 3:5 and a:c = ½ : 3/5 , find the ration of a:b:c.

a:c½:3/55:6

a:b3:5

LCM of 3 and 5 =15 a:b:c = 15:25:18

ANGLES

ao

84o

A = 84O (vertically opposite angle)

ao

84O

A= 840 ( corresponding angles)

ANGLES

ao

840

A = 84o ( ALTERNATE ANGLES)

yo

xo ao

A = xo + yo ( interior angles = exterior angle)

UNITS OF LENGTH

• 1cm = 10mm• 1dm = 10cm • 1m= 100cm• 1km = 1000m

UNITS OF LENGTH

• 1g = 1000mg • 1kg= 1000g • 1 ton = 1000kg

• Capacity = volume

1l = 1000ml 1ml = 1cm2