Contents Unit 1: Prime numbers and factorisation ...

Copyright © Mathematics Mastery 2018 1

Contents

Unit 1: Prime numbers and factorisation ............................................................................................. 2

Section 1.1:Finding factors, multiples and prime numbers ...................................................... 2

Section 1.2: Finding the prime factors of a number using a factor tree ............................. 11

Section 1.3: Determine the highest common factor and lowest common multiple by

prime factorisation ................................................................................................................................. 14

Section 1.4: Problems in context ....................................................................................................... 19

Section 1.5: Using a calculator ............................................................................................................ 22

Unit 2: Add and subtract fractions and mixed numbers…………………………………………......24

Section 2.1: Use equivalent fractions ............................................................................................... 24

Section 2.2: Add and subtract fractions with like denominators ......................................... 25

Section 2.3: Add and subtract fractions with common factors in the denominators ... 26

Section 2.4: Add and subtract fractions with unlike denominators .................................... 28

Section 2.5: Convert between mixed numbers and improper fractions ............................ 31

Section 2.6: Addition and subtraction of improper fractions and mixed numbers ....... 32

Section 2.7: Problems with addition and subtraction fractions ............................................ 35

Unit 3: Positive and negative numbers ................................................................................................ 42

Section 3.1: Represent, order and compare positive and negative numbers .................. 42

Section 3.2: Adding and subtracting negative numbers ........................................................... 46

Section 3.3: Multiplying and dividing negative numbers ........................................................ 53

Section 3.4: Problem solving with negative numbers ............................................................... 58

Unit 4: Sequences, expressions and equations ................................................................................ 63

Section 4.1: Describe and continue number patterns ............................................................... 63

Section 4.2: Generating number patterns ...................................................................................... 66

Section 4.3: Find an expression for the 𝑛th term rule ................................................................ 69

Section 4.4: Generating sequences from the 𝑛th term rule ...................................................... 75

Section 4.5: Algebraic notation .......................................................................................................... 78

Section 4.6: Algebraic expressions ................................................................................................... 80

Section 4.7: Forming algebraic expressions ................................................................................. 83

Section 4.8: Substitution ....................................................................................................................... 85

Section 4.9: Forming algebraic equations ...................................................................................... 86

Section 4.10: Linear equations ........................................................................................................... 88

Section 4.11: Mixed problems ............................................................................................................ 96

2 Copyright © Mathematics Mastery 2018

Unit 1: Prime numbers and factorisation nnn

Section 1.1: Finding factors, multiples and prime numbers

1. Write down the next 5 multiples of:

a) 3 ………………………………………………………………………………………………………………

b) 6 ………………………………………………………………………………………………………………

c) 18 ………………………………………………………………………………………………………………

d) 4.5 ………………………………………………………………………………………………………………

2. Complete the Venn diagram below, using the first 10 multiples for each number.

Concept corner

Numbers in the 4 times table are called ____________________________ of 4.

The first four multiples of 5 are 5, _______, _______, _______.

A ____________________________ of a whole number is

any whole number that divides into it exactly.

The factors of 20 are 1, 2, 4, _______, _______, _______.

Use some of these to fill in the gaps:

factor multiples

two 5 15 20

11 13 10 8

Multiples of 3 Multiples of 4

Common multiples of 3 and 4


3. Write down:

a) a multiple of 6 between 30 and 40 ……………

b) a multiple of 7 between 50 and 60 ……………

4. Write down the first three numbers that are multiples of both of the numbers:

a) 4 and 5 ……………………………………………………………………………………………………….

b) 4 and 6 ……………………………………………………………………………………………………….

c) 4 and 7 ……………………………………………………………………………………………………….

5. Circle the factors of 12 and the factors of 16.

a) Factors of 12

Factors of 16

b) Use your answers to complete the Venn diagram below.

c) What is the highest common factor of 12 and 16? ………………………….

Factors of 12 Factors of 16

Common factors


6. Find all the factors of each of these numbers:

a) 18 ………………………………………………………………………number of factors …………..

b) 25 ………………………………………………………………………number of factors …………..

c) 40 ………………………………………………………………………number of factors …………..

d) 11 ………………………………………………………………………number of factors …………..

e) 36 ………………………………………………………………………number of factors …………..

f) 37 ………………………………………………………………………number of factors …………..

g) 24 ………………………………………………………………………number of factors …………..

h) 1 ………………………………………………………………………number of factors …………..

7. Sort the numbers in question 6 into three groups.

How many factors do prime numbers have? ………………………………………………………….

8. Sort the numbers in question 6 into two groups.

Square numbers always have an ………………………………number of factors.

Exactly

1 factor

Exactly

2 factors

More than 2 factors

Odd number of factors Even number of factors


9.

a) Which of the above represent square numbers? Why?

…………………………………………………………………………………………………………………………

b) Draw diagrams to show 16 and 25 are square numbers.

10. Jay has arranged dots into three different rectangles to represent the factors of 12.

He says that he can only arrange the dots into one rectangle to represent the

factors of 7.

Why is this?

What would the rectangle look like?

4 × 3 = 12

a. b. c. d. e. f. g. h. i.

12 × 1 = 12

6 × 2 = 12


11. List all of the prime numbers up to 20.

………………………………………………………………………………………………………………………………

12. List all of the square numbers up to 100.

………………………………………………………………………………………………………………………………

13. Is 101 a prime number? Explain how you know.

………………………………………………………………………………………………………………………………

14. From the numbers below, choose one number that fits each of these descriptions.

a) A multiple of 3 and 4 ……………………………………………………………………..

b) A square number and an even number ………………………………………………..

c) A factor of 12 and a factor 16 ………………………………………………………………

d) An odd factor of 30 and a multiple of 3 ………………………………………………..

e) A number with exactly 4 factors ………………………………………………………….

f) A multiple of 5 and a factor of 30 …………………………………………………………

g) An odd number and a factor of 36 ………………………………………………………..

h) An even number and a factor of 24 and a multiple of 6 …………………….….

i) A prime number that is two more than a square number ………………………

j) An odd number that is a multiple of 7 …………………………………………………

16 3 21 10 9 4 11 14 12


15. Are the following statements true or false? Justify your answers.

a) All prime numbers are odd numbers True/False

…………………………………………………………………………………………………………………………

b) All odd numbers are prime numbers True/False

…………………………………………………………………………………………………………………………

c) The only even prime number is 2 True/False

…………………………………………………………………………………………………………………………

d) There are six prime numbers less than 10 True/False

…………………………………………………………………………………………………………………………

16. Prime numbers such as 3 and 5 that differ by 2 are called twin primes.

List 5 other pairs of twin primes.


17. A perfect number is a positive integer that is equal to the sum of its factors,

excluding itself.

The smallest perfect number is 6, because the factors of 6 (not including 6) are

1, 2 and 3 which sum to 6 (e.g. 1 + 2 + 3 = 6).

a) Work out the next perfect number.

b) There are three perfect numbers less than 500. What are they?

18. What is the same and what is different between these two calculations?

……………………………………………………………………………………………………………………………...

2 + 2 + 2 = ? 2 × 2 × 2 = ?


19. Fill in the gaps in the calculations below.

a) 5 × 5 = 5

b) 5 × 5 × 5 × 5 = 4

c) 5 = 5 × 5 × 5

d) 5 = 5

e) 5 + 5 + 5 = 3 ×

f) 3 + 3 + 3 + 3 + 3 = 3 ×

g) 54 = 5 5 5 5

h) 6 × 3 = 6 6 6

i) 3 = 37 × 3

j) 6 × 6 = 65

20. Write the following in index form:

a) 3 × 3 × 3 × 3 …………………………………………

b) 8 × 8 × 8 × 8 × 8 …………………………………………

21. Work out the values of:

a) 25 …………………………………………………………………..

b) 73 …………………………………………………………………..

c) 53 …………………………………………………………………..


22. Write the following products in index form:

a) 2 × 2 × 5 × 5 × 5 …………………………………………………………………..

b) 2 × 2 × 7 × 7 × 7 × 3 …………………………………………………………………..

c) 11 × 5 × 11 × 5 × 11 …………………………………………………………………..

d) 3 × 2 × 2 × 5 × 3 × 2 × 3 …………………………………………………………………..

23. Work out the value of:

a) 23 × 32 …………………………………………………………………..

b) 24 × 7 …………………………………………………………………..

c) 22 × 32 × 5 …………………………………………………………………..

d) 32 × 52 × 11 …………………………………………………………………..


Concept corner

Prime factorisation will result in the same expression no matter what order the prime factors are identified. Complete each of the following factor trees to demonstrate this:

Write a calculation for 30 as product of its primes: 30 = ……… × ………… × …………

Section 1.2: Finding the prime factors of a number using a factor tree

1. By drawing at least two factor trees, write the following numbers as the products of their prime factors.

a) 24

Write a calculation for 24 as product of its primes.

24 = ……… × ………… × ………… × …………

Give your answer in index form:

24 = ……… × …………

30

10 4

30

2

30

6 5


b) 48

Write a calculation for 48 as product of its primes and in index form.

48 = ……… × ………… × ………… × ………… × ………… = ………… × …………

c) 192

Hint: Use your answers to parts a) and b)

192 = ……………………………………………………………………………………………………


2. A number is expressed as the product of its prime factors 23 × 3 × 52

What number is it? ………………………………………………………………………………………………...

3. The prime factors of a number are 2, 5 and 11.

What are the three smallest numbers with these prime factors?

4. Which is the smallest number that has four different prime factors?

5. Which of these numbers are perfect squares?

You may use a factor tree for each to help work this out.

a) 256 b) 960


Concept corner

The h ____________________ c _________________ f __________________ of a set of numbers is the

highest number that is a factor of all of the numbers.

The l___________________ c_______________________ m____________________ of a set of numbers is

the lowest number that is a multiple of all of the numbers.

Example 1

Find the highest common factor of 48 and 120.

48 = 𝟐 × 𝟐 × 𝟐 × 2 × 𝟑 = 24 × 3

120 = 𝟐 × 𝟐 × 𝟐 × 𝟑 × 5 = 23 × 3 × 5

So the highest common factor of 48 and 120 is 24.

Example 2

Find the lowest common multiple of 24 and 30.

24 = 2 × 2 × 2 × 3 = 23 × 3

30 = 2 × 3 × 5

So the lowest common multiple of 24 and 30 is 23 × 3 × 5 = 120.

2 × 2 × 2 × 3 = 24

Section 1.3: Determining the highest common factor and lowest common multiple

by prime factorisation

1. a) Write 28 as a product of its prime factors.

2

2 2

2

3

5

48 120

2 2

2 3

5

24 30


b) Fill in the boxes to complete the Venn diagram below.

What is the highest common factor? …………………………………………………

What is the lowest common multiple? …………………………………………………

2. Find the four different pairs of numbers with a common factor of 6 whose prime

factors are 2, 2, 3 and 7.

One answer has been started for you.

Work out the highest common factor and lowest common multiple of each of these pairs of numbers.

2 2

3

7

42

2

2 28

3

5


3. a) Complete the missing values in the Venn diagrams below.

b) Work out the highest common factor for each set.

c) Work out the lowest common multiple for each set.

d) Explain why one is the same and the other is different.

4. Draw Venn diagrams and find the highest common factor and lowest common

multiple of:

a) 36 and 54

2 2 2 3

5 2

3 5


b) 60 and 50

c) 66 and 40.

d) 42 and 90


5. Use factorisation to help you simplify these fractions:

a) 36

120 Hint: highest common factor = 12

b) 48

210 Hint: highest common factor = 6

c) 9

24×

24

108

d) 3

8×

16

24

e) 4

9÷

8

18

f) 𝑎

𝑏÷

𝑎

𝑏𝑐

6. Find the highest common factor and lowest common multiple of:

a) 𝑎2𝑏𝑐 and 𝑎𝑏²𝑐.

b) 9𝑥2𝑦 and 𝑥𝑦

c) 2𝑠𝑡𝑢 and 4𝑡2𝑢𝑣


Section 1.4: Problems in context

1. Rebecca and Sally were each given a piece of ribbon of equal length. Rebecca cut her

ribbon into equal lengths of 8 m, while Sally cut her ribbon into equal lengths of 6 m.

If there was no ribbon leftover, find the shortest possible length of ribbon given to

them.

2. Two buses leave the depot at 8:30 am. The number 13 bus leaves every 15 minutes

and the number 100 bus leaves every 20 minutes. When do they next leave the depot at the same time?


3. Imagine that is the year 3000. Three planets

orbit a star and are lined up as shown in the

diagram. These planets take 8, 9 and 10

Earth months respectively to orbit their

star. In what year will all three planets be

lined up again in the same position?

4. We want to cut the equal sized squares from

this sheet of paper.

What should the length of the sides of the

largest square be so there will not be any

paper left over?

36

cm

24 cm


5. You are asked to form a square by laying

rectangular papers that are 5 cm long and 7 cm

wide without any gaps or overlaps.

a) What is the length of the side of the

smallest square you can make?

b) How many rectangular pieces of paper do

you need to make the smallest square?

7 cm

5 cm


Section 1.5: Using a calculator

1. Use a calculator to find these square roots, giving your answers correct to 2 decimal places:

a) √6 …………… b) √10 ……………

c) √414 …………… d) √40 ……………

2. What are the lengths of the sides of a square which has an area of 49 cm²?

………………

3. A square has an area of 160 cm².

How long are the sides of this square, to the nearest mm? ………………………………………

4. Fill in the boxes below, using two consecutive integers in each question.

a) < √10 < b) < √90 <

5. What is the perimeter of a square with area 196 cm²?

6. Three identical squares are put side by side to form a rectangle. The area of the

rectangle is 147 cm². What are the lengths of sides of the rectangle?

Using a calculator

On my calculator, to find √54 , I press: _____________________________________________

(Check: √54 = 7.35 to 2 decimal places. Did you get this answer?)


Reflections

This space is for you to write your reflections on the whole unit on prime numbers

and factorisation.

You may wish to write about:

Things you’ve learnt

Things you found difficult

Other areas of maths you used in this topic

Topics you need to revisit/revise in the future


Unit 2: Add and subtract fractions and mixed numbers nnn

Section 2.1: Use equivalent fractions

1. Complete the number lines:

Concept corner

Complete the vowel-less words:

F r …... c t …... …... n

V …... n c …... l …... m

D …... n…... m …... n…... t …... r

N …... m …... r…... t …... r

…... q …... …... v …... l …... n t

…... m p r …... p …... r

Use these words to help you name the different parts of a fraction:

3

7


2. Use the number lines, or otherwise, to compare the following fractions by writing

the correct symbol in the box (<, >, =).

1

7

1

8

7

9

5

6

2

3

3

4

Section 2.2: Add and subtract fractions with like denominators

1. The sum of two bricks is equal to the brick above.

Complete the pyramids below:

1

8

1

8

2

8

2

8

2

12

4

12

5

12

1

9

7

9

4

9

4

11

2

11

10

11


Concept corner

Section 2.3: Add and subtract fractions with common factors in the denominators

Complete these calculations and bar models:

1. Calculate:

a) 3

10+

1

20=

b) 3

16+

3

8=

c) 9

24+

1

12=

d) 7

10+ 0.2 =

e) 0.1 + 2

5=

f) 3

20+ 0.13 =

1

2+

1

4=

4+

4 =

4

3

10+

1

5=

10+

10 =

10

3

8+

1

2=

+

=

1

4+

3

8=

+

=


Concept corner

What is the same and what is different about these calculations?

………………………………………………………………………………………………………………………………….

1

2 −

1

4 =

4 −

4 =

4

3

10 −

1

5 =

10 −

10 =

10

5

8

−

1

2 =

−

=

5

8 −

1

4 =

−

=

Complete these calculations and label the bar models:

2. Calculate:

a) 3

10−

1

20=

b) 9

16−

3

8=

c) 11

24−

5

12=

d) 7

10− 0.2 =

e) 0.1 + 2

5−

3

10=

f) 3

20− 0.13 +

1

5=


Concept corner


………………………………………………………………………………………………………………………………….


………………………………………………………………………………………………………………………………….

Section 2.4: Add and subtract fractions with unlike denominators

Complete these calculations and label the bar models:

1. Calculate:

a) 1

4+

1

3=

b) 1

3−

1

4=

c) 1

4+

5

9=

d) 4

5−

1

9=

e) 1

8+

1

6=

f) 1

6−

1

8=

1

2+

1

5=

10+

10 =

10

2

5+

1

2=

10+

10=

1

2−

1

5=

10−

10=

1

2−

2

5=

−

=


Concept corner

Think about this calculation: 𝟏

𝟒+

𝟏

𝟔

Two students have calculated this sum and written their working out in full:

What is the same and what is different between these methods of calculation?

……………………………………………………………………………………………………………………………………

With this in mind would you change your method of working in questions 1e and 1f?

……………………………………………………………………………………………………………………………………

2. Calculate:

a) 1

4+

1

12=

b) 7

12−

5

18=

c) 1

2+

4

5−

3

4=

d) 2

5+

3

10−

7

15=

e) 1

2+

1

6−

1

8=

f) 7

8−

4

5+

1

2=

1

4+

1

6=

1 × 6

4 × 6+

1 × 4

6 × 4

= 6

24+

4

24

=10

24

=5

12

1

4+

1

6=

1 × 3

4 × 3+

1 × 2

6 × 2

= 3

12+

2

12

=5

12


3. Ali adds two fractions and writes down

1

4+

2

5=

5

20+

4

20=

9

40

There are several mistakes in this calculation. Explain where Ali has gone wrong and do the question correctly.

4. A garden has an area of 1

4 of a hectare. The owner buys an extra

2

5 of a hectare of

land.

a) What is the total area of the garden now?

b) How much more land would the owner need to have a garden with an area of

1 hectare?

5. A group of friends went to a coffee shop. 2

5 of them only bought a coffee and another

1

3 of them only bought a cup of tea. The rest only bought a cake.

a) What fraction of the group bought a cake?

b) What is the smallest possible number of people in the group?


Section 2.5: Convert between mixed numbers and improper fractions

1. Change the improper fractions to mixed numbers:

a) 9

4= b)

25

3= c)

48

5=

2. Change the mixed numbers into improper fractions:

a) 81

4= b) 3

4

5 = c) 2

5

3=

3. Put these fractions in descending order:

……………………………………………………………………………………………………………………………….......

Key idea

A fraction greater than 1 can be written in two ways

𝟕

𝟒

𝟏𝟑

𝟒

= = Improper

fraction

Mixed number

17

6

21

4

17

5

14

5

15

6 3

1

4

23

4


Section 2.6: Addition and subtraction of improper fractions and mixed numbers

1. Calculate:

a) 31

7+

2

7=

b) 51

3+

1

9=

c) 41

6−

1

9=

d) 21

3+ 3

1

6=

e) 32

9+ 3

1

3=

f) 31

6+ 2

2

9=

g) 31

3− 2

1

6=

h) 35

6− 2

2

9=

2. The sum of the two bricks is equal to the brick above.


21

10

31

20

103

10

31

12

71

6

1

6

51

4

85

8

2 31

18 4

1

18

77

18


Concept corner

Think about this calculation: 𝟑𝟒𝟓

+ 𝟏𝟑

Two students have calculated this sum and written their working out in full:


……………………………………………………………………………………………………………………………………

Check to see if 42

15 and

62

15 are equal.

……………………………………………………………………………………………………………………………………

3. Complete the following calculations:

a) 22

3+

5

6=

b) 4

5+ 3

7

10=

c) 5

6+ 5

3

4=

d) 54

5+ 2

1

4=

e) 35

9+ 2

5

6=

f) 22

3+ 4

5

7=

34

5+

1

3= 3

12

15+

5

15

= 317

15

= 42

15

34

5+

1

3=

19

5+

1

3

= 57

15+

5

15

=62

15


Concept corner

Think about this calculation: 𝟑𝟏𝟒

− 𝟐𝟑

Two students have calculated this subtraction and written their working out in full:


……………………………………………………………………………………………………………………………………

4. Complete the following calculations:

a) 41

6−

2

3=

b) 31

5−

2

3=

c) 52

3− 2

5

6=

d) 74

9− 2

5

6=

31

4−

2

3= 3

3

12−

8

12

= 3 +3

12−

8

12

= 3 + (−5

12)

= 27

12

31

4−

2

3=

13

4−

2

3

= 39

12−

8

12

=31

12

= 27

12


Section 2.7: Problems with addition and subtraction of fractions

1. Calculate the perimeters of the following shapes, expressing your answer in its simplest term.

Perimeter = …………….…



7

4 cm

Diagrams not

drawn accurately

1

2 m

5

12 m

1

3 m 1

4 m


2. You are given the perimeter of each shape.

Calculate the length of the labelled sides:

…………….…

Perimeter = 9.5 cm

…………….…

Perimeter = 15.3 cm

…………….…

Diagrams not

drawn accurately

Perimeter = 24 cm 147

20 cm

52

5 cm


3. Ian wins £214

million. He gives £ 35

million to charity and £ 18

million to his friends

and family. How much does he have left from the winnings?

4. Which of the following is greater and by how much?

A: The difference between 81

8 and 4

1

3

B: The sum of 11

3 and 2

2

5

5. In a magic square the rows, columns and diagonals all add up to the same number. This is called a ‘magic number’.

Complete the magic square below:

What is the magic number? ………….

2

10

1

14

5

11

5


6. To complete the magic square below let 𝑎 = 2

5, 𝑏 = 3

1

5, 𝑐 = 2

1

3

What is the magic number? ………….

7. In each set below, which is the greatest?

117

8 or 11.66 or

96

8

2

3+

3

4 or 0.75 or

38

20− 1

2 3

5 + 2.43 or

23

5 + 0.17 or 6.25

3

8

7.2 2 1

2 or 1

2

5 + 2.67 or 7.22

8

3

𝑎 − 𝑏 𝑎 − 𝑐 𝑎 + 𝑏 + 𝑐

𝑎 𝑎 − 𝑏 + 𝑐

𝑎 + 𝑐 𝑎 + 𝑏

𝑎 − 𝑐 + 𝑏

𝑎 − 𝑏 − 𝑐


8. On Monday, a shop sold 151

2 kg of sweets.

On Tuesday, the shop sold 5

6 kg more sweets than on Monday.

On Wednesday, the shop sold 11

3 kg fewer sweets than on Tuesday.

The shop had 1

3 kg of sweets leftover.

How many kilograms of sweets did the shop have at the start of the week?

You can use the bar model below to help answer this question.

Monday

Tuesday

Wednesday

Leftover

?

15 1

2 kg

15 1

2 kg

5

6 kg

1

3 kg

11

3 kg


9. Bucket A contained 37

12 litres of water.

Bucket B contained 1 2

3 litres less water than bucket A

Bucket C contained 5

6litres less water than bucket B.

The water in the three buckets was poured into a large container.

How much water is in the container?

Using a bar model or otherwise to help you answer this question.


Reflections

This space is for you to write your reflections on the unit adding and subtracting

fractions.







Concept corner

We can use the number line to compare integers.

Which is greater?

–5 or –2

Write a negative decimal number and a negative fraction.

……………………………………………………………………………………………………………………………….....

Unit 3: Positive and negative numbers

Section 3.1: Represent, order and compare positive and negative numbers

1. Fill in the missing numbers on the thermometer.

Negative integers Positive integers

0°C

10°C

−10°C

5°C

15°C

20°C

25°C

30°C

35°C Put the correct symbol, either < or >,

between these pairs of temperatures.

−10 5

10 −5

−10 −5

−5 −10

5 −10

−5 10

The temperature is at freezing point.

It falls by 10°C, then rises by 6°C.

What is the temperature now?


2. Fill in the missing numbers on the number lines below.

3. Complete each of the following by putting a suitable number in each box.

a) is less than 2

b) is less than −1

c) 3 is less than

d) −5 is less than

e) is greater than −3

f) ……… . is greater than 1

g) 1 is greater than

h) −1 is greater than

4. Put the correct symbol, either < or >, in the circle.

a) 01

4 −1

b) −5 −41

4

c) −4 −1

d) 0 − 1

4

e) −1

2 −

1

4

f) −1

2 −1

−5 – 2 0 3 5

– 8 – 4 0 2

– 1 −1

2 0

1

4 1


5. Mark the number halfway between the two numbers circled on the number line.

6. a) What number is halfway between −6 and 0? ………………………

b) What number is halfway between −6 and 8? ………………………

a) What number is halfway between −6 and −3? ………………………

b) What number is halfway between −20 and −34? ………………………

7. Complete the sentences below:

a) −2 is halfway between ………......... and 0.

b) −2 is halfway between ………......... and 2.

c) −2 is halfway between ………......... and −5.

d) −2 is halfway between ………......... and −20.

8. Look at the information about the pairs of numbers, 𝑥 and 𝑦.

Find 3 values of 𝑥 when:

𝑦 = 1 ………………………

𝑦 = 0 ………………………

𝑦 = −1 ………………………

𝒙 < 𝒚 and 𝒙𝟐 > 𝒚


9. Write two possible addition or subtraction calculations that could be shown by each

number line below:

The first one has been completed for you.

𝟏 + 𝟒 = 𝟓

𝟓 − 𝟒 = 𝟏


Key idea

Section 3.2: Adding and subtracting negative numbers

1. Complete these calculations. 5 + 0 = 5

5 + − 1 =

5 + − 2 =

5 + − 3 =

5 + = 1

5 + = 0

5 + − 6 = 5 + = −5

7 + −5 = 2

6 + = 1

5 + = 0

4 + −5 =

+ −5 = −2

2 + −5 =

+ −5 = −4

+ −5 = −11

3 + 2 = 5

3 + 1 = 4

3 + 0 = 3

3 + −1 = 2

−2 − 2 = −4

−2 − 1 = −3

−2 − 0 = −2

−2 − −1 = −1

How are these representations the same?

How are these representations different?

Remember:

3 + − 1

≡ 3 − 1

−5 + 6 = 1

3 + −2 =

−2 − −2 =

−2 − − 1 ≡ −2

+ 1


2. Calculate:

a) 7 + − 4 =

𝑐) 7 + − 8 =

𝑏) 11 + − 12 =

𝑑) − 11 + − 12 =

3. True or false?

a) − 3 + 8 = 5

c) 4 + −2 = 2

b) 8 + −5 = 3

d) − 1 + −3 = 2

e) −99 + −99 = 0

g) − 1.3 + −2.6 + 1.1 = −5

f) −0.2 + − 1.5 = −2

h) −3

8+ −

1

4+ −

1

12= −

17

24

4. Correct all the incorrect calculations in question 3.



2 −7 −5 −5 −9 −11

−1

3 −3

1

15 −

11

5

−9.4 −21.7 −8.3


6. Complete the calculations:

−3 − 1 = −4

−3 − 0 =

−3 − −1 =

−3 − −2 =

−3 − = 0

−3 − = 1

−3 − −5 =

−3 − −10 =

−4 − −1 = −3

−3 − −1 =

−2 − −1 =

−1 − = 2

0 − = 1

1 − = 12

2 − −1 =

12 − −1 =

7. Calculate:

a) 7 − − 4 =

b) 2 − −1 =

c) −7 − −4 =

d) −2 − −1 =

8. True or false?

a) 3 − 8 = 5

b) 6 = −4 − 2

c) −99 − −99 = 0

d) −1.3 − −2.6 − −1.1 = 2.4

e) 8 − −5 = 3

f) 4 = 1 − −3

g) −0.2 − −1.5 =1.7

h) −3

8− −

1

4− −

1

12=

1

3





11. Fill in the missing numbers:

a) 4 − = 1

b) 4 − = −1

c) 4 + = −1

d) 4 + = 1

e) 4 + + = 1

f) 4 + + + = 1

g) 4 + + + + = 1

h) 4 + + + + = −1

i) 4 + + − + = 1

j) 4 + − + − = −1

−11

−14

−31

−

1

8

−5

6

−17

24


12. Write two numbers that add to 6.

One of the numbers must be positive.

The other number must be negative.

13. Write two numbers that add to −6

One of the numbers must be positive.

The other number must be negative.

14. Complete these calculations:

6 + =

10 + =

10 − =

−10 + =

−10 − =

−6 + =


15. Complete these calculations using only negative numbers:

16. In a magic square the rows, columns and diagonals all add up to the same number.

Complete the following magic squares.

a)

3 −4

0 2

−3

Magic number …………

b)

13

1

−11 10


c)

0.1 −1.7

−0.7

−1.5


d)

1

2 −

5

6 1

1

3

11

2


9 − =

−9 − =


17. If 𝑥 = 2 and y = −5, evaluate

a) 𝑥 + 𝑦 =

b) 𝑥 + 𝑦 + 𝑦 =

c) 𝑥 − 𝑦 =

d) 𝑦 − 𝑥 =

18. If 𝑥 = −2, y = −5 and z = −10, evaluate

a) 𝑥 + 𝑦 =

b) 𝑥 + 𝑦 + 𝑧 =

c) 2𝑥 + 𝑦 =

d) 𝑥 − 𝑦 =

e) 𝑦 − 𝑥 =

f) 3𝑧 − 𝑦 =

19. Complete the table below:

𝒂 17 7 9 6 5 −2

𝒃 12 −5 −6 −6

𝒂 − 𝒃 −11 −3

𝒃 − 𝒂 3 −7

−𝒂 − 𝒃


Section 3.3: Multiplying and dividing negative numbers

1. Complete the calculations:

−7 × 1 = −7

−7 × = −14

−7 × 3 =

−7 × 4 =

−7 × 5 =

−7 × 6 =

−7 × 7 =

−7 × = −56

−7 × 9 =

−7 × 10 =

−7 × 11 =

−7 × 12 =

−6 × −1 = 6

−6 × = 12

−6 × −3 =

6 × −4 =

6 × −5 =

−6 × −6 =

−6 × −7 =

−6 × = −48

−6 × −9 =

6 × −10 =

−6 × 11 =

−6 × −12 =

2. Complete the multiplication table.

× −3 −2 −1 0 1 2 3

3

2

1

0

−1

−2

−3

Shade the positive numbers one colour and the and negative numbers in a different

colour.

What patterns can you see? …………………………………………………………………………………….......


3. Calculate.

a) −12 × 3 =

b) 12 × −3 =

c) −3 × −12 =

d) −3 × −4 × −3 =

4. Complete the multiplication grids:

× 5 1 4

2

2

3

× 5 1 −1

2

4.2

1

2

3

5. True or False?

a) −7 × 3 = −21

b) 6 × −8 = 48

c) −3 × −2 = 3 × 2

d) −4 × 7 = 4 × −7

e) (−5)2 = −5 × −5

f) −8 × −8 = (−82)

g) −7 × 4 × −3 = 7 × −4 × 3

h) −6(2 + 3) = (−6 × 2) + (−6 × 3)

i) −23 = −22 × 2

j) −3 × 3 × −3 × 3 = −32 × 3


Key idea

3 × 2 = 6 −3 × 2 = −6

3 × −2 = −6 −3 × −2 = 6

What

connections

can you see?


Key idea Fact families

Complete the follow:

−𝟑 × 𝟒 = −𝟏𝟐

𝟒 × −𝟑 = −𝟏𝟐 ÷ = −𝟑

−𝟏𝟐 ÷ = 𝟒

−𝟓 × 𝟔 = −𝟑𝟎

6 × = −𝟑𝟎

÷ −𝟓 = 𝟔

−𝟑𝟎 ÷ =

What other facts can you derive from these fact families?

………………………………………………………………………………………………………………………………

…………………………………….

7. Complete these calculations.

24 ÷ −12 = −2 24 ÷ −8 = 24 ÷ −6 =

−24 ÷ 12 = −2 −24 ÷ 8 = −24 ÷ 6 =

−24 ÷ −12 = 2 −24 ÷ −8 = −24 ÷ −6 =

8. What other facts can you derive from the equations below?

Some have been started for you:

9. Calculate.

a) −18 ÷ 3 =

b) 18 ÷ −3 =

c) −18 ÷ −6 =

d) −18 ÷ −9 × −3 =

−6 × 4 = −24 6 × −4 =

−24 ÷ 4 =


10. Complete the multiplication grids.

× 4 −9

−8 18

−3 −12

35 −14

16

× 4 −9

−8 18

−3 −12

35 −14

16

11. True or False?

a) −21 ÷ 3 = −7

b) −6 = −48 ÷ −8

c) 30 ÷ −2 = −15

d) −14 ÷ 7 = 2

e) 24 ÷ −6 = −48 ÷ 12

f) −3 = −63 ÷ −21

g) −𝑎

𝑏= −𝑎 ÷ −𝑏

h) −𝑚𝑛

−𝑚= 𝑛

12. Correct all the incorrect equations in question 11.

Complete the same multiplication

grid a different way.


13. If 𝑥 = 2, y = −5 and z = −10, evaluatethe following.

a) 𝑧 ÷ 𝑦 =

b) 𝑧 ÷ 𝑥 =

c) 𝑥𝑦 =

d) 𝑥𝑦𝑧 =

e) 𝑥𝑦 ÷ 𝑧 =

f) 𝑦2 =

14. Evalute the expressions for below.

a)

b)

𝑛 = −4

𝑛 + 3 =

𝑛 − 3 =

3 − 𝑛 =

3𝑛 − 3 =

3𝑛 + 3 =

3𝑛 =

𝑛 = −4 𝑛2= 𝑛2 + 5 =𝑛2 + 5

3=


15. Evalute the expressions.

Section 3.4: Problem solving with negative numbers

1. True or False?

a) −7 × 3 = −7 + −12 + 2

b) −40 ÷ −8 = −6 + 1

c) 3 × −2 × −4 = −15 − −9

d) −2 − −2 = 3 − −3

e) 0.2 + − 0.8 = 5 × −0.2

f) 4(8 − 11) = −144 ÷ 12

g) −32 = −5 − 4

h) −2

5− −

8

5=

15

5× −

2

5

i) 5

6− −

3

4= 2 −

5

12

j) −72 × −3 = −(−7 − 4)2

2. Correct all the incorrect equations in question 1.

𝑛 = −5

𝑛2 =

1

2𝑛 + 3 =

3(𝑛 −1

2) =

𝑛(𝑛 −3

2) =

3𝑛

2=

3𝑛2

2=


3.

a) What else could go in the box below? (You can use any operation +. −, ÷, × )

……………………

b) Complete the diagram below.

4. Use these cards to complete the diagram below

−𝟏𝟐

6

−𝟑

−𝟑 6

−𝟏𝟐

3

𝟔

−𝟒

1

÷ 3 × −

1

4 −12

12 ÷ − 2 −6


5. Use these cards complete the diagram below

0 −𝟏𝟎

2

7

10

−𝟑

𝟏

𝟐

1

4

−2 +8 × −2 ÷ −4


6. Use the numbers in the brackets to complete each equation:

a) + × = 3

(−5, −2 and − 7)

b) + × = − 17

(5, −2 and − 7)

c) + × = − 37

(5, −2 and − 7)

d) + × =33

(−5, −2 and − 7)

e) + × = − 19

(−5, 2 and − 7)

f) + × =9

(−5, −2 and − 7)

g) ( + ) ÷ = − 6

(−5, 2 and − 7)

h) ( + ) ÷ =1

(−5, 2 and − 7)

i) ( + ) ÷ = − 1

(5, 2 and − 7)


Reflections

This space is for you to write your reflections in the unit on negative numbers.







Unit 4: Sequences, expressions and equations

Section 4.1: Describe and continue number patterns

1. Here is part of a number grid.

a) What number is in the square below the number 18? …………………

b) What number is in the square below the number 15? …………………

c) Here is another part of the same grid.

What are the missing numbers?

2. Fill in the missing numbers.

Term to term rule: Add 6

11 17 23

35


3. Fill in the missing numbers and write the term-to-term rule for each sequence.

a)

b)

c)

d)

e)

f)

8 16 24

Term-to-term rule:

18 26 34

4 12 20

22 14 6

3 2.2 1.4

−2.2 −1.4 −0.6

Term-to-term rule:

Term-to-term rule:

Term-to-term rule:

Term-to-term rule:

Term-to-term rule:


Key idea

A sequences is ascending if the terms get bigger.

A sequence is descending if the terms get smaller.

Is each the sequence below ascending, descending or neither?

Ascending Descending Neither

2, 3, 4, 5, 6, 7, …

1

2,2

4,3

6,4

8,

5

10,

6

12, …

−2, −3, − 4, − 5, − 6, …

4. Fill in the missing numbers and state whether the sequence is ascending or

descending.

a)

b)

c)

1

2

1

3

1

4

1

5

Ascending or descending?

6

13

7

12

8

11

9

10



3

2

4

3

5

4

6

5


Section 4.2: Generating number patterns

1. The term-to-term rule in a number sequence is add 7.

a) Use this rule to write the missing numbers in the sequence.

b) The term-to-term rule in a different number sequence is double, then add 3.

Use this rule to write the missing numbers in the sequence.

2. Each term-to-term rule below generates a sequence.

Write the next two numbers in each sequence.

a)

b)

c)

3. The term-to-term rule of the sequence is multiply by 2 and add 4,

The third term of the sequence is 6.

Work out the missing terms in this sequence.

Rule: Add 6 to the previous number

4 10 16 22 ………. ……….

Rule: Double the previous number

4 8 16 32 ………. ………. Rule: Multiply the previous number by 3 then add 2

2 8 26 80 ………. ……….

8 15

11 25 53

6


Concept corner

Unless a rule is stated, we can’t be sure how a sequence continues and ‘spotting a

pattern’ may lead to incorrect conclusions.

For example, a sequence starting with 1, 2, 3 … may continue

Write a rule for this sequence ………………..………………………………..………………………

Or…


Can you continue this sequence in a different way?


4. Continue each sequence in two different ways and write a rule for each sequence.

a)

b)

1 2 3 4 5 6

1 2 3 5 8 13

1 2 3

Rule:

1 2 4

Rule:

1 2 4


5. In this sequence the same number is added each time.

Use this rule to write the missing numbers in the sequence.

6. The rule for this sequence is to add the same number each time.

Use this rule to write the missing numbers in the sequence





3 15

Term-to-term rule:

3 15

3 −9

3 −21

Term-to-term rule:

Term-to-term rule:

Term to term rule:


Section 4.3: Find an expression for the 𝒏th term rule

Concept corner

Matchsticks are arranged to make this pattern.

Sketch the next shape in the pattern.

Fill in the blanks below.

To get to the 3rd term, I start with 3 and add on 2, two times.

To get to the 4th term, I start with _______ and add on 2, __________times.



To get to the 𝑛th term, I start with 3 and add on 2, 𝑛 – 1 times.

1st term 2nd term 3rd term

To get the 𝑛th term 3 + 2(𝑛 – 1)

Expand and simplify = 3 + 2𝑛 – 2

= 2𝑛 + 1

The 𝒏th term rule for this sequence is 2𝒏 + 1.

Term number 𝒏

𝟏 𝟐 𝟑 𝟒 𝟓 𝟔

Number of matches

3 5

+ 2


1.

a) Sketch the next shape in the pattern.

b) Complete the table below.

Term number (𝒏)

𝟏 𝟐 𝟑 𝟒 𝟓 𝟔

Number of matches

3 6

c) What is the 𝒏th term rule? ………………………………………………………………………….

d) Use the 𝒏th term rule to calculate how many matchsticks will be needed to make

a pattern with 50 triangles.

+ 3



2.



Term number (𝒏)

𝟏 𝟐 𝟑 𝟒 𝟓 𝟔

Number of matches


d) Use the 𝒏th term rule to calculate how many matchsticks will be needed to make

a pattern with 50 squares.

1st term

2nd term 3rd term


3.



Term number (𝒏)

𝟏 𝟐 𝟑 𝟒 𝟓 𝟔

Number of matches


a) Use the 𝒏th term rule to calculate how many matchsticks will be needed to make

a pattern with 50 triangles.

4. Draw the first three terms of the matchstick pattern with the 𝒏th term rule 3𝒏 + 3.



5. Work out the 𝑛th term rule for each of the matchstick patterns below:

a)

b)

c)

d) What is the same and what is different about each of these sequences?





6. Draw lines to match each number sequence to its 𝑛th term rule.

7. Work out the formula for the 𝑛th term rule for each of the following sequences:

a) 8, 13, 18, 23, 28, …

b) 13, 8, 3, 2, −7, …

c) −13, −8, −3, 2, 7, …

d) −2, −7, −12, − 17 …

e) −8, −13, −18, −23, − 28, …

4𝑛

4𝑛 + 1

4𝑛 + 2

4𝑛 − 1

5, 9, 13, 17, …

3, 7, 11, 15, ..

4, 8, 12, 16, …

6, 10, 14, 18, …


Concept corner

You can make ‘huts’ with matchsticks.

The 𝒏th term rule to find out how many matchsticks you need is 𝟒𝒏 + 𝟏.

Use the rule to find out how many matches you need to make 9 huts.

Section 4.4: Generating sequences from the 𝒏th term rule

1. Write the first five terms of a sequence whose 𝑛th term rule is:

a) 5𝑛 + 4

b) 5𝑛 − 4

c) 4 − 5𝑛

d) −4 − 5𝑛

Use the rule to find out how many matches you need to make 9 huts.

Use the rule to generate the first 6 terms in the sequence.

5, ………., ……….., ……….., ……….., ………..,

If I had 81 matchsticks how many huts

could I make?


I used the 𝑛th term rule to see how many

huts I could make with 65 matchsticks.

4𝑛 + 1 = 𝟔𝟓

4𝑛 = 64

𝑛 = 16

If I had 65 matchsticks, I could make

16 huts.


2. Draw lines to match each number sequence to its 𝑛th term rule.

3. To find the 𝑛th triangle number, you can use this rule:

a) Work out the 8th triangle number.

b) Now work out the 100th triangle number.

𝑛th triangle number = 𝑛

2(𝑛 + 1)

4𝑛

8𝑛 − 4

8 − 4𝑛

4𝑛 + 4

4, 12, 20, 28, …

8, 12, 16, 20, …

4, 8, 12, 16, …

4, 0, −4, −8


4. Here are the 𝑛th term rules for three different sequences.

The first three terms of each sequence are 1, 2 and 4.

What is the 4th term of the each sequence?

a) Sequence A

b) Sequence B

c) Sequence C

1 2 4

2(𝑛−1) 𝑛2 − 𝑛 + 2

2

𝑛(𝑛2 − 3𝑛 + 8)

6

1 2 4

1 2 4

Sequence A Sequence C Sequence B


Section 4.5: 𝐀𝐥𝐠𝐞𝐛𝐫𝐚𝐢𝐜 𝐧𝐨𝐭𝐚𝐭𝐢𝐨𝐧

1. Match each statement to the correct expression.

2. Match each expression with the equivalent expression below.

2𝑦 + 𝑦 2𝑦 − 𝑦 2𝑦 × 𝑦 2𝑦 ÷ 𝑦

3𝑦 2 𝑦 2𝑦2 2𝑦 3𝑦2

Add 5 to 𝑔

Subtract 5 from 𝑔

Multiply 𝑔 by 5

Divide 𝑔 by 5

Divide 𝟓 by 𝑔

Subtract 𝑔 from 𝟓

5

𝑔 − 5

𝑔 + 5

5𝑔

𝑔

5

5 − 𝑔

𝑔5

𝑔

5

𝑔


3. Write the correct operation (+, −, ÷, ×) in these statements to make them true.

𝑥 𝑥 ≡ 2𝑥

𝑥 𝑥 ≡ 1

𝑥 𝑥 ≡ 0

𝑥 𝑥 ≡ 𝑥2

4. Write the following functions algebraically using a variable of your choice.

For example, we can write “multiply by 6, then add 8” as 6𝑦 + 8.

a) Multiply by 6, then add 11

b) Multiply by 15, then subtract 7

c) Divide by 8, then subtract 11

d) Add 9, then multiply by 4

e) Subtract 12, then multiply by 1.8

f) Add 7, then divide by 5

g) Multiply by 3, then subtract 5, then divide by 7

h) Double, then subtract from 11


Section 4.6: 𝐀𝐥𝐠𝐞𝐛𝐫𝐚𝐢𝐜 𝐞𝐱𝐩𝐫𝐞𝐬𝐬𝐢𝐨𝐧𝐬

1. Fill in the blanks in these statements.

a) 𝑦 + 𝑦 + 𝑦 + 𝑦 + 𝑦 ≡ _____________

b) 𝑐 + 𝑐 − 𝑐 + 𝑐 − 𝑐 ≡ ___________

c) 𝑤 + 𝑤 − 𝑤 _________________ + 𝑔 + 𝑔 + 𝑔 + 𝑔 + 𝑔 ≡ 2𝑤 + 5𝑔

d) 3𝑏 + 4𝑑 ≡ 𝑏 + 𝑏 + 𝑏 − 𝑏 + 𝑏 + ___________________

e) c + 𝑥2 + 𝑥2 − 𝑐 − 𝑐 − 𝑐 + _____________________ ≡ _________ + 5𝑥2

f) 𝑟 × 𝑟 + 𝑠 + s + s + s + s ≡ _______________

g) 𝑥2 + 2𝑦 + 2𝑧 ≡ ____________________________________________

Concept corner

Complete the vowel-less words:

…… x p r …… s s …… …… n

……q …… …… v …… l……n t

L……k…… t……r m s

S…… m p l …… f …… ……d

V……r…… …… b l ……s

S…… m ……

Use these words to help you fill in the blanks below:

_____________________ can be simplified by collecting _____________________________:

𝑔 + 𝑔 + 𝑔 ≡ 3𝑔 𝑟 + 𝑟 + 3 + 2 ≡ 2𝑟 + 5 𝑡 + 𝑡 + 𝑡 + 𝑞 + 𝑞 ≡ 3𝑡 + 2𝑞

The identically equal (≡) symbol means that these expressions are ___________________________

Equivalent expressions will always have the _____________________ numerical value




3. Use the signs + and – to complete the following

a) 3𝑥 7𝑦 8𝑥 5𝑦 ≡ 11𝑥 12𝑦

b) 3𝑥 7𝑦 8𝑥 5𝑦 ≡ 11𝑥 2𝑦

c) 3𝑥 7𝑦 8𝑥 5𝑦 ≡ −5𝑥 2𝑦

d) 3𝑥 7𝑦 8𝑥 5𝑦 ≡ −5𝑥 12𝑦

4. True or false?

a) 𝑎 + 𝑎 + 𝑎 ≡ 3𝑎

b) 3𝑎 ≡ 𝑎 × 𝑎 × 𝑎

c) 3 + 𝑎 ≡ 3𝑎

d) 7𝑏 – 𝑏 ≡ 7

e) 7𝑏 – 7 ≡ 𝑏

f) 7𝑏 – 𝑏 ≡ 6𝑏

10𝑡

23𝑡

53𝑡

3𝑘 𝑘 𝑘

3𝑓 + 𝑔

6𝑓 + 6𝑔

10𝑓 + 8𝑔

8𝑎 + 9𝑏 2𝑎 − 3𝑏 3𝑎 − 5𝑏


5. Here are the rules for completing the algebra grid.

a)

b)

c)

2𝑥 + 3

2𝑥 3

6𝑥

This is the sum of the middle row

This is the product of the middle row

𝑥 2

𝑥 𝑥

𝑥2 1

4𝑥 + 5

4𝑥

8𝑥

4𝑥

2𝑥

4𝑥

3𝑥

9𝑥

3𝑥

3𝑥2

3𝑥

−9𝑥


Section 4.7: Forming algebraic 𝐞𝐱𝐩𝐫𝐞𝐬𝐬𝐢𝐨𝐧𝐬

1. Write down a simplified expression for the perimeter and area of each of these

shapes.

Shape Perimeter Area a)

b)

c)

d)

2. Write an expression for the unshaded area in the shape below.


3. A rectangle has area 𝟏𝟐𝒂𝟐 and perimeter 𝟏𝟒𝒂.

What are the dimensions of this rectangle?

4. Use your knowledge of angle facts to write equations to represent the information in

each diagram below.

a)

b)

c)

d)


Section 4.8: Substitution

1. Look at this equation 𝒙 + 𝒚 = 𝟏𝟏

Write three different solutions to this equation.

𝒙 = … … … …. and 𝒚 = … … … ….

𝒙 = … … … …. and 𝒚 = … … … ….

𝒙 = … … … …. and 𝒚 = … … … ….

2. Evaluate each expression below.

Circle the expressions with the highest and lowest values.

a) When 𝒂 = 𝟏.

b) When 𝒂 = 𝟓.

c) When 𝒂 = −𝟓.

3. Complete the missing values in this table.

𝒙 𝒙 + 𝟏 𝟒𝒙 𝟒𝒙 − 𝟐 𝟒(𝒙 − 𝟐)

3 10

9

64

2 + 𝑎 2𝑎 𝑎2 𝑎

2 10 − 𝑎

2 + 𝑎 2𝑎 𝑎2 𝑎

2 10 − 𝑎

2 + 𝑎 2𝑎 𝑎2 𝑎

2 10 − 𝑎

Further practice

For more questions on substitution visit the following pages:

Fractions – page 38

Negative numbers – pages 52, 57 and 58

Generating sequences – pages 75, 76, and 77.


Section 4.9: Forming algebraic equations

1. Decide if each of the statements below are true or false for this bar model:

a) 𝑎 = 𝑏 + 𝑐

b) 𝑏 = 𝑎 + 𝑐

c) 𝑐 = 𝑎 − 𝑏

d) 𝑐 = 𝑏 − 𝑎

e) 𝑎 − 𝑐 = 𝑏

f) 𝑐 − 𝑎 = 𝑏

g) 𝑐 + 𝑏 = 𝑎

2. Write down fact families for these bar models: a) b)

3. Which of the equations below are equivalent to 𝟔𝒙 + 𝟗 = 𝟐𝟒

𝒂

𝑏 𝑐

𝒙

𝑦 𝒛

𝒑

𝑞 𝒓 𝑞

2𝑥 + 3 = 8 24 − 6𝑥 = 9 6𝑥 = 33 𝑥 = 1.5 48 − 12𝑥 = 18


4. Use the equation in the middle to find the missing numbers

2𝑥 + 7 = 37.85

2𝑥 + 8 = __________________________ 2𝑥 + 3 = __________________________

4𝑥 + 16 = __________________________

2𝑥 + 10 = __________________________

2𝑥 = __________________________

4𝑥 + 14 = __________________________

4𝑥 = __________________________


Section 4.10: Linear equations

1. Complete the equations for the bar models below.

Use these to work out the missing value of 𝒂.

a)

b)

c)

Concept corner

Fill in the gaps to complete the equations:

𝒂 + 13 = 59 59 − 13 = 𝒂

+ 𝒂 = 59 59 − 𝒂 =

Which equation will help you work out the value of 𝒂?

59

𝒂 13

55

𝒂 27

75.3

𝒂

55

𝒂 27 𝒂

𝑎 + 27 = 55 − = 27

+ 𝒂 = 55 55 − = 𝑎

𝑎 + 𝑎 + 𝑎 = 3 × = 75.3

÷ 𝟑 = 𝑎 ÷ 𝒂 = 3

𝑎 + 27 + 𝑎 = 2𝑎 + 27 =

55 – 27 = 2 = 55−27

2

𝒂 𝒂

𝒂 = ____

𝒂 = ____

𝒂 = ____


2. Solve these equations.

a) 𝑥 + 6.2 = 9.4

b) 𝑚 – 2.3 = 8.8

c) 3𝑐 = 18.6

d) 9.8 = 2𝑝 + 1.6

Concept corner

3𝑥 + 4𝑥 ≡ 7𝑥 is an identity; this is true for all values of 𝑥.

3𝑥 + 7 = 11.5 is a linear equation there is only one solution, a value of 𝑥 for which it is

true.

Can you see how we get

from one step to the next?

𝑥 𝑥 𝑥 7

11.5

𝑥 6.2

9.4

𝑚 2.3

8.8

𝑐

18.6

𝑐 𝑐

𝑝

9.8

𝑝 1.6


3. Solve the equations below.

a) 6 = 9𝑧 − 12

d) − 3.2𝑑 = 12.8

b) 6𝑦 = −24

e) 𝑥 − 1.6 = −3.6

c) 4𝑡 + 1 = −5

f) 2 – 4𝑑 = 8

4. Circle the odd one out.

a) 2𝑥 = 30 2𝑥 + 8 = 22 14 = 2𝑥 𝑥 = 7

b) 3𝑥 − 8 = 25 8 − 3𝑥 = 25 3𝑥 = 33 −33 = −3𝑥

c) 𝑡 = −1 8𝑡 = −8 8𝑡 + 3 = −5

8𝑡 = −2

d) 4𝑑 = −6 4 – 8𝑑 = 16 4𝑑 = 6 2 – 4𝑑 = 8


Form and solve equations for questions 5 to 11

5. I think of a number, add 14.

The answer is 47.

6. I think of a number, take it away from 47.

The answer is 11.

7. I think of a number, multiply it by 3.

The answer is 48.

8. I think of a number, divide it by 4.

The answer is 14.

9. I think of a number, multiply it by 5 and then add 6.

The answer is 41.

10. Four times my number taken away from 31 leaves 11.

11. Three times my number taken away from 52 leaves 24.

? 14

47

? 11

47


Concept corner

Think about this equation: 𝟑(𝒙 + 𝟕) = 𝟒𝟖

Two students have solved this equation and written their working out in full:

Which method do you prefer? Why?

……………………………………………………………………………………………………………………………………

12. Solve these equations. a) 3(𝑥 + 8) = 51

b) 16 = 2(5 + 2𝑥)

c) 2(𝑥 − 2) = 9.8

d) 12.4 = 4(3𝑥 + 1)

3(𝑥 + 7) = 48

𝑥 + 7 = 16

𝑥 = 9

3(𝑥 + 7) = 48

3𝑥 + 21 = 48

3𝑥 = 27

𝑥 = 9

𝑥 7

𝑥 7

𝑥 7

48

𝑥 7 𝑥 7 𝑥 7

𝑥 7 𝑥 𝑥 7 7

48

27

𝑥 7

16


13. Form and solve an equation to help you solve this problem.

I think of a number, subtract 10 and then multiply by 2. The answer is 30.

What is my number?

14. Complete the pyramids and find the values of 𝒂 and 𝒃.

If the values in the bottom row were in a different order, would you get a different

number at the top of the pyramid?

Use the pyramids below to support your reasons.

12 5 𝒂

33

12 𝒃 5

37


15. Solve these equations

a) 𝑥

6.2=3.3

b) 𝑦

4.2+ 12.4 = 13.3

c) 2 (𝑓

4+ 8.3) = 20.8

d) 2

5(𝑡 + 3.2 ) = 2.8

e) 4(𝑦 – 1) + 5(ℎ𝑦 + 1) = 100 f) 4(ℎ – 1) − 5(ℎ + 1) = 0

16. Circle the odd one out.

a) 5(2𝑥 − 3) = 40 10𝑥 − 15 = 40 (2𝑥 − 3) = 8 10𝑥 − 15 = 25

b) 4(5 − 6𝑥) = −40 (5 − 6𝑥) = −10 20 − 6𝑥 = −40 20 − 24𝑥 = −40

c) 7𝑦 + 11 = 98 7𝑦 = 52 7𝑦 + 46 = 98 2(𝑦 + 3) + 5(𝑦 + 8) = 98

d) 7𝑧 + 21 − 5𝑧 + 10 = 123 7𝑧 + 21 − 5𝑧 − 10 = 123 7(𝑧 + 3) − 5(𝑧 − 2) = 123


17. Form and solve equations to calculate the missing lengths.

a)

b)

18. Form and solve the equations to calculate the missing angles.

a)

b)

2𝑡

5 cm

𝑥 cm

2𝑥 cm

10 cm

Perimeter: 39 cm

(𝑦 − 5) cm

𝑦 cm

(𝑦 + 1) cm

(𝑦 − 5) cm

Perimeter: 23 cm

3𝑡

5𝑡

2𝑡 = ______ 3𝑡 = ______ 5𝑡 = ______

𝑥 5𝑥 30°

The diagrams are not

drawn accurately.


Section 4.11: Mixed problems

1. Look at this sequence of numbers 2, 5, 8, 11, …

Is 33 a number in this sequence? Give a reason for your answer.

2. A sequence is made of black and white tiles.

a) Each term in the sequence has 1 black tile and each new term has more white tiles.

How many white tiles are added each time?

b) What is the 𝒏th term rule for the total number of tiles?

c) How many tiles will there be in the 9th term of this sequence?

d) What term of the sequence had 64 tiles?



3. Each column of the table is added together is equal to the total column.

E.g. 𝑎 + 𝑎 + 𝑎 = 12.9

Total

𝑎 𝑎 𝑎 12.9

𝑎 𝑏 𝑏 10.7

𝑎 𝑏 𝑐 13.8

Work out the values of 𝒂, 𝒃 and 𝒄.

4. Write an algebraic equation to represent each row, column and diagonal.

Use your equations to help you find the missing totals.

11.9

13.7

12.2

12.6 ? 12.2 ?


5. The diagram represents a number sequence.

The sequence continues by adding 𝑦 each time.

Bart says that the 𝑛th term rule is 𝒙 + 𝒏𝒚.

Bart is wrong.

What is the 𝑛th term of the sequence?

6.

a) Write an expression, in terms of 𝑥, for the total weight of the packages.

b) The packages weigh 18 kilograms altogether. By forming and solving an equation,

find the weight of each of the packages.

𝒙 𝒙 + 𝟑 𝟑𝒙

𝒙 1st term

𝒙 𝒚 + 2nd term

𝒙 𝒚 + 3rd term 𝒚 +


7. Which of these sequences will contain the number 100?

2, 5, 8, 11, 14 …

1, 4, 7, 10, 13 …

4, 10, 16, 22 …

5, 11, 17, 23 …

Give reasons for your answers.

8. Look at the triangle below.

Work out the value of 𝑝.

Diagram not drawn

to scale.


9. The 𝑛th term rule of a sequence is 𝟒𝒏 + 𝟏.

The 𝑛th term rule of another sequence is 𝟑𝒏 – 𝟏.

Work out numbers that that are in both sequences and between 20 and 50.

10. Triangle ABC is an isosceles triangle.

Calculate the value of 𝑡.

Diagram not drawn

to scale.


11. Work out the value of each shape.

Each shape has an integer value.

48

2

= = = = =


Reflections

This space is for you to write your reflections on this unit sequences, expressions and

equations.








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