Cambridge Primary Mathematics Workbook 6 (Cambridge Primary Maths) [2 ed.] 1108746330, 9781108746335

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Cambridge Primary Mathematics Packed with activities, including drawing angles, completing sequences and working out ratios, these workbooks help your students practise what they have learnt. Specific exercises develop thinking and working mathematically skills. Focus, Practice and Challenge exercises provide clear progression through each topic, helping learners see what they’ve achieved. Ideal for use in the classroom or for homework.

CAMBRIDGE

• Activities take an active learning approach to help learners apply their knowledge to new contexts • Three-tiered exercises in every unit get progressively more challenging to help students see and track their own learning • Varied question types keep learners interested • Write-in for ease of use • Answers for all activities can be found in the accompanying teacher’s resource

Primary Mathematics Workbook 6

For more information on how to access and use your digital resource, please see inside front cover.

This resource is endorsed by Cambridge Assessment International Education

✓ P rovides learner support as part of a set of

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✓ Developed by subject experts ✓ For Cambridge schools worldwide

Mary Wood, Emma Low, Greg Byrd & Lynn Byrd

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Second edition

Digital access

CAMBRIDGE

Primary Mathematics Workbook 6 Mary Wood, Emma Low, Greg Byrd & Lynn Byrd

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108746335 © Cambridge University Press 2021 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Second edition 2021 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Printed in Dubai by Oriental Press. A catalogue record for this publication is available from the British Library ISBN 978-1-108-74633-5 Paperback with Digital Access (1 Year) Additional resources for this publication at www.cambridge.org/9781108746335 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. NOTICE TO TEACHERS IN THE UK It is illegal to reproduce any part of this work in material form (including photocopying and electronic storage) except under the following circumstances: (i)

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Contents

Contents How to use this book

5

Thinking and Working Mathemetically

6

1

The number system

8

1.1 1.2

Place value Rounding decimal numbers

8 13

2

Numbers and sequences

18

2.1 2.2 2.3

Counting and sequences Special numbers Common multiples and factors

18 23 27

3 Averages

32

3.1

Mode, median, mean and range

32

4

Addition and subtraction (1)

38

4.1 4.2

Positive and negative integers Using letters to represent numbers

38 43

5

2D shapes48

5.1 Quadrilaterals48 5.2 Circles53 5.3 Rotational symmetry60

6

Fractions and percentages65

6.1 Understanding fractions 6.2 Percentages 6.3 Equivalence and comparison

7

65 69 73

Exploring measures 77

7.1 Rectangles and triangles 7.2 Time

77 83

8

Addition and subtraction (2)89

8.1 8.2

Adding and subtracting decimal numbers Adding and subtracting fractions

89 94

3

Contents

9 Probability98 9.1

Describing and predicting likelihood

98

10 Multiplication and division (1)107 10.1 Multiplication 10.2 Division 10.3 Tests of divisibility

107 111 114

11 3D shapes119 11.1 Shapes and nets 11.2 Capacity and volume

119 127

12 Ratio and proportion136 12.1 Ratio 12.2 Direct proportion

136 140

13 Angles146 13.1 Measuring and drawing angles 13.2 Angles in a triangle

146 154

14 Multiplication and division (2)160 14.1 Multiplying and dividing fractions 14.2 Multiplying decimals 14.3 Dividing decimals

160 164 167

15 Data171 15.1 Bar charts, dot plots, waffle diagrams and pie charts 15.2 Frequency diagrams, line graphs and scatter graphs

171 182

16 The laws of arithmetic194 16.1 The laws of arithmetic

194

17 Transformations199 17.1 Coordinates and translations 17.2 Reflections 17.3 Rotations

199 208 215

Acknowledgements219

4

How to use this book

How to use this book This workbook provides questions for you to practise what you have learned in class. There is a unit to match each unit in your Learner’s Book. Each exercise is divided into three parts: • Focus: these questions help you to master the basics • Practice: these questions help you to become more confident in using what you have learned • Challenge: these questions will make you think more deeply. Each exercise is divided into three parts. You might not need to work on all of them. Your teacher will tell you which parts to do. You will also find these features: Important words that you will use.

Step-by-step examples showing a way to solve a problem.

There are often many different ways to solve a problem. These questions will help you to develop your skills of thinking and working mathematically.

5

Thinking and Working Mathematically

Thinking and Working Mathematically There are some important skills that you will develop as you learn mathematics.

Specialising is when I give an example of something that fits a rule or pattern. Characterising is when I explain how a group of things are the same. Generalising is when I explain a rule or pattern. Classifying is when I put things into groups.

6

Thinking and Working Mathematically

Critiquing is when I think about what is good and what could be better in my work or someone else’s work.

Improving is when I try to make my work better.

Conjecturing is when I think of an idea or a question to develop my understanding.

Convincing is when I explain my thinking to someone else, to help them understand.

7

1 The number system 1.1 Place value Worked example 1

compose  decimal point  decompose

Paulo is thinking of a number.

digit  hundredths  place value

He says, ‘If I divide my number by 10 and then by 100, the answer is 0.375.’

regroup  tenths  thousandths

What number is Paulo thinking of? To find Paulo’s number, reverse the operations.

0.375 × 100 × 10 100

10

1

0.1

0.01

0.001

0

3

7

5

5

0

3

7

3

7

5

0.375 × 100 × 10 = 375 Answer: Paulo is thinking of 375.

8

You could replace × 100 × 10 by × 1000. × 100 × 10

1.1 Place value

Exercise 1.1 Focus 1 Draw a ring around the expression that is equivalent to 0.67. 6 10

+

7 60            10 10 6 10

            +

+

7 100

7 60           100 100

+

70 100

2 What does the digit 5 in 3.065 represent? 3 Magda regroups 56.079 in different ways but two of her answers are wrong. Which answers are wrong? A: 5607 tenths + 9 thousandths B: 56 ones and 79 thousandths C: 56 + 0.79 D: 50 + 6.079 E: 50 + 6 + 0.07 + 0.009

9

1 The number system

4 Write the operations to complete these multiplication and division loops. 3.7

0.034

÷ 10

37

0.37

34

0.34

0.98

0.098



98

5 Complete the place value diagram.

90

91.969

+

+

+

0.06

+

6 Write the number six tenths, four hundredths and five thousandths as a decimal. Practice 7 Complete the table to show what the digits in the number 47.506 represent. 4 5 6 7

10

tens

1.1 Place value

8 Find the missing numbers. a 5.6 × 100 =

b 0.88 × 1000 =

c 41.28 × 10 =

d 670 ÷ 1000 =

e 191 ÷ 100 =

f

6.3 ÷ 10 =

9 Draw a ring around the expression that is equivalent to 4.063. A: 4 + 0.6 + 0.3

B: 4 + 0.6 + 0.03

C: 4 + 0.06 + 0.03

D: 4 + 0.06 + 0.003

10 Petra puts some numbers into a function machine. in

× 1000

out



Complete the table to show her results. in

out

1.5

1500 937

16.24 490 0.07 11 Write the decimal number that is represented by

–4 – 20 –

7 100

–

6 1000

–

9 10



11

1 The number system

Challenge 12 Ingrid says, ‘I can multiply by 100 by adding two zeros.’ Explain why Ingrid is wrong.

13 Filipe multiplies a number by 10, then again by 10 and again by 10. His answer is 7.

What number did he start with?

14 Four students Anton, Ben, Kasinda and Anya each think of a number.

The numbers are 45, 4.5, 0.45 and 0.045.



Use these clues to work out which number each student is thinking of. • Ben’s number is a thousand times smaller than Kasinda’s number. • Anton’s number is ten times smaller than Kasinda’s number. • Anya’s number is ten times bigger than Ben’s number.

12



Anton’s number is



Kasinda’s number is



Ben’s number is

Anya’s number is

1.2 Rounding decimal numbers

15 Leila says, ‘The number represented in the place value grid is the largest number that can be made with nine counters.’

Do you agree?



Explain your reasoning. 10s

1 10

1s

1 100

1 1000





1.2 Rounding decimal numbers Worked example 2

nearest

Neve has four number cards. 0.25

1.25

2.25

round 3.25

She chooses two cards. She adds the numbers on the cards together. She rounds the result to the nearest whole number. Her answer is 4. Which two cards did she choose?

13

1 The number system

Continued 1.25 and 2.25

Find two numbers that add to 3.5 as 3.5 rounds to 4

or 0.25 and 3.25

You could choose 1.25 and 2.25 or 0.25 and 3.25

You are specialising when you choose two numbers and check if the total rounds to 4.

Exercise 1.2 Focus 1 Draw lines to show each number rounded to the nearest tenth.

The first one has been done for you. to the nearest tenth

8.52

8.77

8.35



8.3 8.4 8.5 8.6 8.7 8.8

2 Draw a ring around all the numbers which equal 10 when rounded to the nearest whole number.

10.53 10.5 10.35 9.55



10.05 9.5 9.05 9.35

3 a Round 7.81 to the nearest tenth. b Round 7.81 to the nearest whole number.

14

Tip Remember the numbers could be less than 10 or more than 10.

1.2 Rounding decimal numbers

4 Complete the table. Number

Number rounded to the nearest tenth

Number rounded to the nearest whole number

3.78 4.45 3.55 4.04 Practice 5 Choose the largest number from the list that gives 100 when rounded to the nearest whole number.

100.55    99.99    100.9    100.45



    100.5     99.5    99.9

6 Use each of the digits 9, 4, 1 and 2 once to make the decimal number closest to 20. 7 Pedro has four number cards.

0.45

1.45

2.45

3.45

He chooses two cards. He adds the numbers on the cards together. He rounds the result to the nearest whole number. His answer is 5. Which two cards did he choose? and

15

1 The number system

8 Huan is thinking of a number. She rounds it to the nearest whole number. She says, ‘My number is the largest number with 2 decimal places that rounds to 10.’

What number is Huan thinking of?

Challenge 9 Here are eight numbers.

3.36



Use the clues to identify one of the numbers.

2.71

4.03

3.34

3.29

3.15

2.93

3.44

• The number rounds to 3 to the nearest whole number. • The tenths digit is odd. • The hundredths digit is even. • The number rounds to 3.3 to the nearest tenth. 10 Write the letters of all the numbers that round to 10.5 to the nearest tenth.



16

What word is spelt out? A

B

C

D

E

F

G

H

I

10.81

10.56

10.32

10.65

10.44

10.57

10.44

10.43

19.8

J

K

L

M

N

O

P

Q

R

10.48

10.71

10.51

10.58

10.55

9.24

10.59

10.42

10.57

S

T

U

V

W

X

Y

Z

10.44

10.58

10.54

16.25

10.05

10.35

10.46

10.41

1.2 Rounding decimal numbers

11 Stefan says, ‘When I round 16.51 and 17.49 to the nearest whole number, the answer is the same. When I round 16.51 and 17.49 to the nearest tenth, the difference between the answers is one.’

Explain why Stefan is correct.

12 Draw lines from the containers to the circle that shows each measurement rounded to the nearest litre.

10.5 litres

9459 millilitres

8 litres

10400 millilitres

7.65 litres

9 litres

8.82 litres

9.91 litres

10 litres

8100 millilitres

11011 millilitres

11 litres

11.1 litres

9.49 litres



17

2 Numbers and ­sequences 2.1 Counting and sequences Worked example 1

position  position-to-term rule

Write a sequence of five terms with steps of constant size that has first term 1 and 2 second term 1  .

term  term-to-term rule

3

2 3

1 ,

1,

1 3

2 ,

3,

3

2 3 2 3

The step size is the difference between the 1st and 2nd terms. It is  . You could use a number line to help you with the count.

1

18

1

13

2

13

2

1

23

2

23

3

1

33

2

33

4

2.1 Counting and sequences

Exercise 2.1 Focus 1 Here is a rocket made of seven shapes.



Magda draws a sequence using the rockets.

1



2

3

She records information about the sequence in a table. Position

1

2

3

Term (number of shapes)

28

a Complete the table. b What is the term-to-term rule for the sequence? c What is the position-to-term rule for the sequence? d What is the 25th term in the sequence?

19

2 Numbers and ­sequences

2 Felipe counts up in steps of 0.3 starting at 4.

Write the first five terms of Felipe’s sequence.

3 Write the next two terms in each sequence. a 1.4, b

1 , 2

c 0,

1.5,

1.6, 1 2

1,

1 ,

–0.3,

1.7, 1 2

2,

–0.6,

 ,

2 , –0.9,

 , –1.2,

 ,

4 a Find the position-to-term rule for the numbers in this table. Position

Term

1

9

2

18

3

27

4

36

b What is the 10th term of the sequence 9, 18, 27 …? 5 $1 = 100 cents a Complete the table. $ (position)

1

2

5

10

100

cents (term) b What is the position-to-term rule for the sequence 100, 200, 300 …?

20

2.1 Counting and sequences

Practice 6 Given the first term and the term-to-term rule, write down the first six terms of each sequence. Then find the position-to-term rule and the 50th term. a First term: 9, term-to-term rule: add 9

First six terms:



Position-to-term rule:



50th term:

b First term: 11, term-to-term rule: add 11

First six terms:



Position-to-term rule:



50th term:

7 Safia counts back in steps of 0.5 starting at 2.9. a What is the 5th number in her sequence? b What is the 10th number in her sequence? 8 a Follow the instructions in the flow diagram to generate a sequence.

Start

Write down 9

Is the answer more than 100?

Add 9

Write down the answer

YES

Stop

NO

b What is the position-to-term rule for the sequence?

c Imagine the sequence continues forever.







What is the 60th term in the sequence?

21

2 Numbers and ­sequences

9 Kiki counts in steps of 0.03 starting at 3.26. What are the next three numbers in her count?

3.26, 3.29, 3.32,

,

,

,

10 Write a sequence of five terms with steps of constant size which has 2 5

a first term of 1 and a second term of 1 . Challenge 11 a Write the first five numbers in a sequence that starts at 42 and has a term-to-term rule of add 0.15 b What is the 10th term in the sequence?   

 

12 A sequence has a position-to-term rule of multiply by 6. Complete the table. Position

Term

1 2 30 36 72 13 The numbers in this sequence increase by equal amounts each time. a Write the three missing numbers.

10,

,

,

, 42

b What is the term-to-term rule for the sequence?  c Salma says, ‘The position-to-term rule is multiply by 8. Is she correct? Explain your answer.

22









2.2 Special numbers

14 Ahmed counts back in steps of 3 4

1 4

3 4

3 4

starting at 3  .



He counts 3 , 3, 2



Which of these numbers does Ahmed say?



4

1 2

1

1 2

–4

….

1 4

–6

3 4

–8

1 4



2.2 Special numbers Worked example 2

cube number

Use each of the digits 1, 3, 4, 5, 6 and 8 once to make the following 2-digit numbers.

square number

A square number A cube number A multiple of 5

A square number 6

8

The only possible cube number is 64. 5 must be in the ones place for the multiple of 5.

4

A cube number

5

A multiple of 5

1

A square number

That leaves 1, 3 and 8. Use 8 and 1 to make a square number and put 3 in the tens place in the multiple of 5.

6

4

A cube number

3

5

A multiple of 5

Place these digits first.

23

2 Numbers and ­sequences

Exercise 2.2 Focus 1 Calculate. a 12 =

b 53 =

 

c 92 =

 

 

d 13 =

2 What is the sum of the third square number and the fifth square number?

 

3 What is the difference between the tenth square number and the fourth square number?

 

4 Draw a ring around the expressions that are equal to 6².

6×2



2×2×2×2×2×2

6×6

6+6 6+6+6+6+6+6

5 Annie and Heidi play a game of ‘What’s my number?’



Annie says

Heidi replies

Is the number less than 50?

No

Is the number more than 100?

No

Is the number a cube number?

Yes

What is the number?



 

Practice 6 A number is squared and then 2 is added.

The answer is 6.



What is the number?



24

 

 

2.2 Special numbers

7 Calculate. a 12 × 1 =

 

b 5 × 52 =

 

c 3 × 32 =

 

d 42 × 4 =

 

8 Vincent makes a sequence using patterns of rectangular bricks. a Draw the next pattern in the sequence.



1

2

3

4

b Complete the table. Shape

1

2

Number of bricks

1

4

3

4

5

c How many bricks are needed for shape 10? How do you know? 9 Write a number between 0 and 100 in each space on the Carroll diagram. There are lots of possible answers. Cube number

Not a cube number

Even number Not an even number

25

2 Numbers and ­sequences

10 Write each number in the correct place on the Venn diagram.

1

8

9

10

25

square numbers

27

50

64

cube numbers

Challenge 11 Find two 2-digit square numbers that have a sum of 130. +



= 130

12 Draw a ring around all the square numbers in this list.

13

23

33

43

53

13 Emma uses small cubes to make a larger cube. She uses 16 cubes to make the base of her cube.

How many small cubes does Emma use to make the larger cube? How do you know?

14 Put these values in order starting with the smallest.

52

23

33

32

15 Use each of the digits 2, 3, 4, 6, 7 and 8 once to make these numbers.



26

A square number

A square number

A cube number

A cube number

2.3 Common multiples and factors

2.3 Common multiples and factors Worked example 3 Write these numbers in the correct place on the Venn diagram.

common  common factor

      1

factor  multiple  multiple

2

3

4

factors of 20

5

6

7

factors of 24

What is special about the numbers in the shaded area? 1, 2 and 4 are common factors of 20 and 24

factors of 20 are: 1, 2, 4, 5

factors of 20

5

factors of 24 are: 1, 2, 3, 4, 6

2 4

Make sure you include every number in the diagram. You could tick each number as you place it. 7 is not a factor of either 20 or 24

factors of 24

1

Tip

3 6 7

The numbers in the shaded area are common factors of 20 and 24.

27

2 Numbers and ­sequences

Exercise 2.3 Focus 1 The multiples of 9 are shaded on the hundred square.



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

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

a Draw a ring around all the multiples of 5. b List the common multiples of 5 and 9. 2 Sofia is thinking of a number.

My number is a multiple of 2. My number is a multiple of 7.

28

2.3 Common multiples and factors



Tick the number that Sofia could be thinking of. 9

27

28

72

3 Find all the common factors of these numbers. a 6 and 8 b 8 and 12 4 Complete the sentence.

Every number with a factor of 10 must also have factors of



 ,

  and

 

Practice 5 Here are four labels.

multiplies of 2

 

multiples of 7

 

not a multiple of 2

 

not a multiple of 7

Write each label in the correct place on the Carroll diagram.

28       56

12         48

      63 35

55         47

6 Faisal is thinking of a number. He says, ‘My number is a multiple of 6.’ What three other numbers must Faisal’s number be a multiple of?

29

2 Numbers and ­sequences

7 Here is a Venn diagram for sorting numbers.

Write each number in the correct place on the diagram.



8   9   10   11   12 multiples of 2 multiples of 4

8 Look at this set of numbers.

13   18   21   36   45 a Which two numbers are factors of 90? b Which two numbers are multiples of 6 and 9?

Challenge 9 Write these numbers in the correct place on the Venn diagram.

1   2   3   4   5   6   7   8   9

factors of 30



30

factors of 24

2.3 Common multiples and factors



What is special about the numbers in the shaded area?

10 A light flashes every 4 minutes and a bell rings every 5 minutes.

The light flashes and the bell rings at the same time. How long will it be until this happens again?

11 Omar packs boxes of mangoes and boxes of peaches.

Each box contains the same number of fruits.

Mangoes Organic & Fresh

goes Mananic & Fresh Org



Omar packs 56 mangoes and 49 peaches.



Ahmed says, ‘There will be 8 pieces of fruit in each box.’



Hassan says, ‘There will be 7 pieces of fruit in each box.’



Who is correct? Explain your answer.

Peaches Organic & Fresh

12 Cakes are sold in packs of eight.

Mr Mason wants to buy enough cakes to share equally between six people with no cakes left over.



What is the smallest number of packs he can buy?



Show your working.



31

3

Averages

3.1 Mode, median, mean and range Worked example 1

average  mean  median

What is the range of these ages?

mode  range

43, 54, 67, 22, 43, 18, 19, 61, 59 The highest age is 67. The lowest age is 18.

Find the highest and lowest ages.

67 – 18 = 49

Subtract the lowest from the highest.

The range is 49 years.

Remember to write the range with the correct units.

Exercise 3.1 Focus 1 Fill in the boxes to work out the mean of the following numbers: a 7, 3 and 2

+



÷3=



+

The mean is

=

 .

b 10, 4, 7, 4, 5

+



÷5=



32

The mean is

+

+

 .

+

=

3.1 Mode, median, mean and range

2 Work out the range of these masses. a 2 kg, 5 kg, 11 kg, 2 kg, 10 kg, 9 kg

highest mass





lowest mass

–



range

=

b 150 g, 103 g, 130 g, 127 g, 144 g

highest mass







lowest mass –

range

=

3 Draw lines to match the descriptions to the set of data.

The range is 5.

5, 6, 5, 7, 8



The mode is 5.

5, 3, 4, 9, 8



The median is 5.

2, 6, 4, 7, 4



The mean is 5.

5, 6, 1, 6, 7

Practice 4 Jenny and Carrie took a spelling test each week. These are the scores from 8 tests. Week

1

2

3

4

5

6

7

8

Jenny

9

10

7

3

19

15

12

13

Carrie

17

7

5

11

12

7

15

6

a Work out Jenny’s mean score and work out Carrie’s mean score. b Work out the range of Jenny’s scores and the range of Carrie’s scores.

33

3 Averages

c Tick the true descriptions of Jenny and Carrie’s scores.

Jenny has a higher range of scores, so she scored higher than Carrie.     



Carrie’s mean average is lower than Jenny, so Jenny scored better than Carrie in every test.



Jenny’s average score was higher, but her scores were less consistent.  



Carrie’s range is lower, so her scores were less spread out. Carrie’s average score was lower than Jenny’s.

5 Erik and Halima recorded how many minutes they practised playing the guitar for one week. Here are their times in minutes. Day

1

2

3

4

5

6

7

Erik

6

5

6

8

5

5

7

Halima

9

8

2

9

8

4

9

a Work out the mean average time each person practised for. b Work out the range of Erik’s scores and the range of Halima’s scores. c Write two sentences to describe the amount of time Erik and Halima practised for.

34

3.1 Mode, median, mean and range

6 14



15

16

17

18

14

15

16

17

18

19

19

Make sets of six numbers from these cards that have these mean, mode and median averages: a mean 16, mode 16, median 16

 

 

 

 

 

b mean 17, mode 18, median 17.5

 

 

 

 

 

c mean 16, modes 15 & 17 (bimodal), median 16

 

 

 

 

 

Challenge 7 The mean has been calculated for each set of numbers below. One number in each set is hidden. Work out the missing number. a The mean is 6.

7 9 6

b The mean is 10.

14 11 9 7

c The mean is 15.

11 19 18 13

d The mean is 19.

16 31 4 7 23

e The mean is 51.

47 63 38 49

35

3 Averages

8 Five children have worked out the mode, median and range of their heights, weights and ages. They have recorded them in this table.







Mode

Median

Range

Height

135 cm

132 cm

16 cm

Weight

33 kg

33 kg

16 kg

Age

10 years & 10 months

11 years & 5 months

10 months

Find a possible solution for the heights, weights and ages of the five children and record it here. Child 1

Child 2

Height:

Height:

Weight:

Weight:

Age:

Child 4

Height:

Height:

Weight:

Weight:

Age:

Height: Weight:

36

Age:

Child 3

Child 5



  

Age:

  

Age:

3.1 Mode, median, mean and range

9 These are the times in seconds of two runners in six 100 m races.

Runner 1: 12.7, 10.4, 11.4, 10.8, 12.2, 10.9



Runner 2: 12.5, 11.9, 10.3, 11.6, 10.8, 11.9 a Find the mean and range for each runner. b Give reasons for who is the better runner.

37

4 Addition and subtraction (1) 4.1 Positive and negative integers Worked example 1

integer  negative number

The temperature in Tallinn is –1 °C and in Moscow it is –8 °C. What is the difference between these two temperatures? temperature in Moscow

temperature in Tallinn difference 7

–8

–7

–6

–5

–4

–3

–2

–1

positive number

• Draw a number line. • Mark the temperatures. • Count the number of degrees between the two marks.

Difference = 7 °C

Exercise 4.1 Focus 1 The temperature at 8 a.m. is –2 °C. By midday it is 5° warmer. What is the temperature at midday?

38

Tip Remember you can draw a number line to help you with these questions.

4.1 Positive and negative integers

2 Use the number line to help you answer these questions. +3

+3

–15

a –15 – 3 =

b

+ 3 = –15

3 Work out the difference between each pair of numbers. a 6 and –2

b –3 and –5

c –4 and –8

d –5 and 3

e –6 and –1

f

0 and –2

Practice 4 The table shows the number of rhinos in the world.

The Black rhino has made a comeback from the brink of extinction. Rhino

Number

Black rhino

About 5000

Greater one-horned rhino

More than 3500

Javan rhino

56 – 68

Sumatran rhino

80

White rhino

More than 20 000

Source WWF 2020



Use the information in the table to write an estimate of the total number of rhinos in the world.



39

4 Addition and subtraction (1)

5 How many more people lived in Tokyo than in New York in 2015? City

Population in 2015

Tokyo

9 273 000

New York

8 582 000

6 The temperature is –15 °C. a The temperature rises by 6 °C. What is the new temperature? b The original temperature falls by 6 °C. What is the new temperature? 7 At a ski resort, the morning temperature was –11 °C.

In the afternoon, the temperature was 5 °C.



What was the difference in temperature between the morning and the afternoon?



40

4.1 Positive and negative integers

8 The table shows the temperatures in some cities and the difference in their temperature from London on one day. Complete the table. City

Difference in temperature from London

Temperature (°C)

London

–1

Moscow

24 degrees colder

New York

10 degrees colder

Oslo

13 degrees colder

–25

Rio de Janeiro

26

9 Ola wants to find the answer to 1999 + 1476.

Tick (✓) all the calculations that will give the same answer.



2000 + 1477



2000 + 1475



2005 + 1470



2005 + 1500



2005 + 1400

Challenge 10 Petra is thinking of a number. She adds 4896 to her number, then subtracts 5846. She gets the answer 9481. What number is Petra thinking of? 11 Meera says, ‘I can work out 79 999 – 19 999 in my head.’ Explain how Meera could do the calculation mentally. Work out the answer to the calculation.



41

4 Addition and subtraction (1)

12 Here are ten number cards. –7



2

0

–5

–3

3

9

–8

–4

Choose one card to complete each number sentence. –7



+

=

–4

–5

–

3

=

13 In some countries, people who have been married for many years have special anniversaries. Number of years married

Special anniversary

25

Silver

40

Ruby

50

Golden

60

Diamond

a Mandy and Derek were married in 1972.

In what year was their ruby anniversary?

b Neve and Sean had their diamond anniversary in 2021.

In what year was their silver anniversary?

14 The difference between two numbers is 3.

One number is –2.



What could the other number be?



Find two different answers.



42

–2

4.2 Using letters to represent numbers

4.2 Using letters to represent numbers constant

variable

Worked example 2 a and b each represent a number between 1 and 9 inclusive. Aba knows that a + 4 = b Write all the values Aba can use to make the statement true. a = 1 and b = 5

Work systematically.

a = 2 and b = 6

Start with a = 1:

a = 3 and b = 7

1+4=5

a = 4 and b = 8

so the value of b is 5

a = 5 and b = 9

When a = 5, b = 9 9 is the largest possible number, so you have found all the possible answers.

Exercise 4.2 Focus 1 Hamda plays a board game using a dice. She uses the instructions together with her dice score to work out how many spaces she moves.

d represents the dice score.



For example: Score

Instruction

Spaces moved

d+4

9 spaces

43

4 Addition and subtraction (1)



Work out how many spaces Hamda moves. Score

Instruction

Spaces moved

a 5+d b 3–d c d–2

2 Mira has 10 more bottles of soda than Noura.

SODA

SODA

SODA

SODA

SODA

SODA

SODA

SODA

SODA

a Complete the table where m represents the number of bottles that Mira has and n represents the number of bottles that Noura has. m

15

n

5

11 2

21 16

b Write a number sentence linking m, n and 10.

44

SODA

4.2 Using letters to represent numbers

3 Olaf and Pierre have 23 toy cars altogether.

Olaf has x toy cars and Pierre has y toy cars. a Complete this table to show the number of toy cars each boy has. x (number of toy cars Olaf has)

7

11

y (number of toy cars Pierre has)

4

14 18

b Write a number sentence linking x, y and 23. Practice 4 The diagram shows a right angle divided into two smaller angles.

50° a



Calculate the size of angle a.



a=

 °

5 There are x kiwi fruits and y oranges in a bowl.

Meng knows that x + y = 7.



Write three different pairs of values for x and y.



45

4 Addition and subtraction (1)

6 x and y each represent a number that is a multiple of 5.

x + y = 50



Write all the possible values of x and y.



One is done for you. x

5

y

45

Challenge 7 The perimeter (p) of a regular pentagon is the sum of the lengths of the sides. b



p=b+b+b+b+b



The perimeter of a regular pentagon is 40 cm.



What is the value of b?

8 The perimeter (p) of a square and a regular pentagon is the same. b a



46

p=a+a+a+a

p=b+b+b+b+b

If the perimeter of each shape is 20 centimetres, what is the value of a and b?

4.2 Using letters to represent numbers

9 a If □ = 7 and ○ = 5 what is the value of □ + ○ + ○? b If a = 7 and y = 5 what is the value of a + b + b? c What is the same and what is different about these two questions?

47

5

2D shapes

5.1 Quadrilaterals bisect  decompose Worked example 1

diagonal  justify

I am a quadrilateral. All my sides are equal in length. None of my angles are 90°. I have two pairs of equal angles. What shape am I?

parallel  trapezia

Square or rhombus …

All sides are equal in length.

Cannot be square …

No angles are 90°.

Must be a rhombus.

Two pairs of equal angles.

Exercise 5.1

kite rhombus square

Focus

parallelogram trapezium

1 Name each of these special quadrilaterals. All the names are in the box.

a

48



b





d

e







rectangle  isosceles trapezium

c



f



g

5.1 Quadrilaterals

2 Complete these properties of a kite. There is a diagram to help you.

a It has

pairs of equal sides.

b It has

pair of equal angles.

c The diagonals cross each other at d It has

 °.

line of symmetry.

3 Complete these properties of a rhombus. There are some diagrams to help you. a It has

equal sides.

b It has

pairs of equal angles.

c It has

pairs of parallel sides.

d The diagonals bisect each other at e It has

°.

lines of symmetry.

49

5 2D shapes

Practice 4 Write down the name of the shape being described. a I am a quadrilateral. All my sides are different lengths. Two of my sides are parallel.

I am a

 .

b I am a quadrilateral. All my sides meet at right angles.

My diagonals bisect each other, but not at 90°. I am a

 .

c I am a quadrilateral. I have two pairs of parallel sides, two pairs of equal sides and two pairs of equal angles.

None of my angles are 90°. I am a

 .

5 Jake draws this rhombus and kite. He labels the lines that make the shapes a, b, c, d and e, f, g and h. He draws the shapes so that b is parallel to h, and the angles marked x are the same size. e a b



X

d c

h

g



Write true or false for each of these statements. Justify your answer. i

b is parallel to d

ii h is parallel to f

50

f

X

5.1 Quadrilaterals

iii d is parallel to h iv a is parallel to e 6 Draw a diagram to show how an isosceles trapezium can tessellate.

Challenge 7 a Describe the similarities between a rectangle and a parallelogram. b Describe the differences between an isosceles trapezium and a kite.

51

5 2D shapes

8 Put the shapes a to f through this classification flow chart and write down the letter where each shape comes out. For example, when you start with the square, you end at the letter H. Start Yes Yes

One pair of equal angles?

G

No

Diagonals meet at 90°?

Yes

No

Yes

All angles 90°?

No

H

Two lines of symmetry?

J

No

I

No

One pair of parallel sides?

L

K

a Square



b Rectangle

c Rhombus



d Parallelogram

e Kite



f

Isosceles trapezium

9 A, B and C are three points shown on this grid. D is another point on the grid. y 10 9 8 7 6 5 4 3 2 1 0



A

B

C

0 1 2 3 4 5 6 7 8 9 10

x

a When D is at (7, 4) is quadrilateral ABDC a square? Explain your answer.

52

Yes

5.2 Circles

b Point D moves so that quadrilateral ABCD is a parallelogram.

What are the coordinates of point D? 

c Point D moves so that quadrilateral ABDC is a kite.

Write down two possible sets of coordinates for the point D. 

 

5.2 Circles Worked example 1

centre  circumference

Label the parts of this circle.

compasses  diameter radius

Circumference is the perimeter.

radius circumference

Centre is in the middle. Radius is the distance from the centre to the circumference.

centre diameter

Diameter is the distance across the circle, going through the centre.

53

5 2D shapes

Exercise 5.2 Focus 1 This is how Tami labelled the parts of a circle. radius

circumference

diameter

centre



Explain the mistakes she has made.

2 Measure the radius of each of these circles. a





radius =

b



54



radius =

mm

cm

5.2 Circles

3 Draw a circle with a radius of a 3 cm

b 40 mm

Practice 4 Draw a circle with a radius of a 3.7 cm

b 52 mm

55

5 2D shapes

5 Write true or false for each of these statements. a A radius of 5 cm is the same as a diameter of 10 cm. b A diameter of 6 cm is the same as a radius of 12 cm. c A radius of 70 mm is the same as a radius of 7 cm. d A radius of 45 mm is the same as a diameter of 9 cm. 6 a Draw a dot and label the point C. Make sure there is about 5 cm of space above, below, to the left and to the right of your point. b Draw the set of points that are exactly 4.2 cm from the point C.



56

5.2 Circles

Challenge 7 a Draw a circle with radius 7 cm. Label the circle A.



57

5 2D shapes

b Draw a circle with radius 3 cm, inside circle A so that it touches circle A. Label the circle B. Your diagram should look something like this. A

B



B

A

  or  

c With a ruler, accurately measure the distance between the centre of circle A and the centre of circle B. d What do you notice about your answer to part c and the radii measurements of circles A and B? e Draw two more circles that touch inside. Choose your own radii measurements. Measure the distance between the centres of your two circles. What do you notice? f

Complete this general rule:



The distance between the centres of two touching circles that touch



inside is the same as the

 .

8 Zara wants to draw a pattern made of squares inside circles like this.



58

This is the method she uses to draw one of the squares in a circle.

Step 1: Draw a square. Step 2: Guess where the centre of the square is and mark a dot. Put the point of the compasses on this dot and open the compasses so that the pencil is on a corner of the square. Step 3: Draw a circle. If the corners of the square don’t touch the circle, rub the circle out and try again with a different centre point!

5.2 Circles

a Try to draw a square in a circle using Zara’s method. What do you think of her method?

b Can you improve on her method? If you can think of a better method, write it down.

59

5 2D shapes

5.3 Rotational symmetry Worked example 3

order

Use tracing paper to work out the order of rotational symmetry of a rectangle.

rotational symmetry

Step 1: Trace the shape.

Step 2: Put your pencil on the centre of the shape.

Step 3: Turn the tracing paper one full turn and count the number of times the shape fits on itself. Start

Once

The rectangle fits on itself twice, so it has order 2.

60

Twice

5.3 Rotational symmetry

Exercise 5.3 Focus 1 Use tracing paper to work out the order of rotational symmetry of these shapes.

a



d



b

c



e



f





2 Match each shape to its order of rotational symmetry.

a







i

Order 4

b



ii Order 2

c



d

iii Order 1

iv Order 3

3 a Draw the line of symmetry on to the triangle.

b Write down the order of rotational symmetry of the triangle.

61

5 2D shapes

Practice 4 Write down the order of rotational symmetry of these shapes. a

    



b

  

c

    

d

   

5 Write down the order of rotational symmetry of these patterns.

a

 



b



 

c

 

6 Write down the order of rotational symmetry of these road signs. a

    



b

   



c

      

Challenge 7 Write the letter of each shape in the correct space. Shape A has been done for you. Number of lines of symmetry 0

1

2

3

4

1 Order of rotational symmetry

2 3 4

62

A

5.3 Rotational symmetry

A



B

C

D

E

F

8 Mali is making a pattern from grey and white squares. This is what she has drawn so far.

a On this copy shade in one more square so that the pattern has order 2 rotational symmetry.



63

5 2D shapes

b On this copy shade in five more squares so that the pattern has order 4 rotational symmetry.

c On this copy shade in seven more squares so that the pattern has order 2 rotational symmetry.

9 Sadik has these nine squares.



64

He wants to arrange the squares to form a square pattern with two lines of symmetry and rotational symmetry order 2. Show two ways that he can do this.

6 Fractions and percentages 6.1 Understanding fractions Worked example 1

denominator  improper fraction

Parveen is thinking of a number.

mixed number   numerator

She says, ‘Two-thirds of my number is 30.’

operator  proper fraction

What number is Parveen thinking of? 1 3

of the number is 30 ÷ 2 = 15

1 3

is half of

2 3

3 3

of the number is 15 × 3 = 45

3 3

is three times

1 3

Parveen is thinking of 45.

Exercise 6.1 Focus 1 Represent these divisions as fractions. a 5 divided by 8



c 8 divided by 7

b 4 divided by 3

d 7 divided by 10



65

6 Fractions and percentages

2 a Show one way that five children can share 2 pizzas equally between them.





How much pizza does each child get?

b Show one way that two children can share 5 pizzas equally between them.





How much pizza does each child get?

3 What is

3 2

of 16?

4 Would you rather be given

66

1 2

of $18 or

1 4

of $40? Explain your decision.

6.1 Understanding fractions

Practice 5 Work out the answer to each question to help you find your way through the maze. start

5 2

of 16

48

7 3

40

5 3

of 18

of 12

63

5 2

27

6

6 5

30

7 6

of 9

of 15

55

24

90

14

4 3

of 15

of 22

7 4

of 16

28

20

end

6 Brian reads

1 3

of a 15-page book.

3 4



Carlos reads



Who reads more pages?



Explain how you know.

of an 8-page book.



67

6 Fractions and percentages

7 Complete this table to show fractions of 36. Fraction

1 4

Amount

9

3 4

5 4

7 4

9 4

11 4

8 Find the missing numbers. 4 3

of 24 =

3 2

of 24 =

7 2

of 24 =

24

8 3

Challenge 9 Which is bigger:

3 4

of 24 =

of 32 or

4 3

of 18?

Explain how you know.

10 a Mandy is thinking of a number.

She says, ‘Five-thirds of my number is 45.’



What number is Mandy thinking of?

b Ollie is thinking of a different number.

68



He says, ‘Ten-ninths of my number is 90.’



What number is Ollie thinking of?

6.2 Percentages

11 Use each of the numbers 2, 3, 4 and 5 once to complete these statements.

of 6 = 4

of 12 = 15



6.2 Percentages Worked example 2

percentage

Leo has two apple trees in his garden. He labels them A and B.

per cent operator

A

B

Tree A produces 40 kg of apples. Tree B produces 10% more than tree A. How many kilograms of apples does tree B produce? 10% of 40 =

1 10

of 40 = 4

Find 10% of 40 and add it to 40 to find the mass of the apples produced by tree B.

40 + 4 = 44 Tree B produces 44 kg of apples.

69

6 Fractions and percentages

Exercise 6.2 Focus 1 Find 10% of the quantity, then use your answer to find 20%, 30% and so on. Write the answers under the percentages on the number line. 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

$200

2 Find 10% of these quantities. a 600

b 90 cm  

c $40

d 170 kg  

3 Join each box to the correct amount. Amount 50% of 40

10 12 14

10% of 120

16 18

100% of 16

Practice 4 a Find 10% of 80 b Find 5% of 80 c Find 15% of 80

70

20

6.2 Percentages

5 Here is a grid of ten squares.



What percentage of the grid is shaded?

6 Complete these percentage diagrams. a

10

%

10 75%

0%

80

%

50

%

25

%

10

75%

%

10

60

30

0%

5%

%

6

50

b

20%

80

7 Haibo makes 250 grams of fruit and nut mix.

15% of the mix is raisins. 25% of the mix is cranberries. The rest is almonds. How many grams of almonds does Haibo use?



71

6 Fractions and percentages

Challenge 8 Huan has two plum trees A and B in her garden.

A



B



Tree A produces 30 kg of plums.



Tree B produces 10% less than tree A.



How many kilograms of plums does Huan get from both trees together?

9 75% of a number is 48.

What is the number?



10 50% of children in a sports club go swimming.



72

Tip The answer is not 36.

6.3 Equivalence and comparison



50% of the children who go swimming also dive.



Five children swim and dive.



How many children are in the sports club?



children

6.3 Equivalence and comparison Worked example 3

equivalent fraction

Write these numbers in order of size, starting with the smallest. 2 5

0.2       25%    25% = 0.25 3 10 2 5

4 10

3    0.23 10

Write the fractions and percentages as equivalent decimals.

= 0.3

=

simplest form  simplify

= 0.4

In order: 0.2, 0.23, 0.25, 0.3, 0.4 0.2, 0.23, 25%,

3 2 , 10 5

Write the decimals in order starting with the smallest.

Write the original values in order.

73

6 Fractions and percentages

Exercise 6.3 Focus 1 Write these fractions in their simplest form. a

5 10



b

8 20



c

9 12

2 What fraction of the shape is shaded?



Write your answer in its simplest form.

3 Write these prices in order starting with the smallest.

$4.07  74 cents  $4.70  $0.47  $7.40

1 4

4 Which numbers are equivalent to  ?

0.5  

25   0.75  0.25 100

Practice 5 Write these numbers in their simplest form. a 13 

74

3   15

b 5 

14   35

c 10 

36   45

6.3 Equivalence and comparison

6 Use the symbols or = to make these statements correct.

70%

0.65    60%



23%

1 0.7 5

1 4

0.06    25% 4 0.3 5

2 5

7 Here are three statements about fractions and percentages.

Tick (✓) the statements that are true.



Cross (✗) the statements that are not true and write the correct statement. a

3 5

b

7 100

c

9 10

is equal to 35% is equal to 7%

is equal to 9%

8 Which two cards make the number sentence correct? 6 8



0.35



3 6

39%

>

0.5

70%

1 2



Challenge 9 Omar and Hassan do the same test.

Omar scores 70 out of 80. Hassan scores 70%.



Who has the higher score? Explain how you know.



75

6 Fractions and percentages

10 Write these numbers in order of size, starting with the largest.

4 5

0.7     0.82  75%  

13 20

11 Find two pairs of equivalent fractions from this list. Circle the fraction in its simplest form in each pair.

16 10 9 6 4 2                20 15 12 10 5 3

12 Circle the smaller number in each box.









76

1 2

1 1.2

1 3

1 1.3

1 4

1 1.4

1 5

1 1.5

7 Exploring measures 7.1 Rectangles and triangles Worked example 1

area

Estimate the area of this triangle.

Centimetre squared paper. 10 whole squares are covered by the triangle.

Count whole squares that are covered by the triangle.

There are no squares that are less than half covered.

Do not count squares that are less than half covered.

There are 5 half squares covered by 1 the triangle, that makes 2 whole 2 squares covered.

Pair up squares that are half covered to make whole squares.

I estimate that the area of the 1 triangle is 12  cm2.

Write your estimate using units of area. The squares are centimetre squares so we use cm2.

2

77

7 Exploring measures

Exercise 7.1 Focus 1 Multiply the width of the rectangle by its length to calculate each area.

Write your answer using units of area.

9 km 8 cm

7m

9 km 4m



          



3 cm

   

2 Circle the rectangles that have been divided into two equal pieces.



78

7.1 Rectangles and triangles

3 a What is the area of this rectangle?

3 cm

4 cm



b Divide the area of the rectangle by 2 to find the area of one of the triangles. What is the area of the triangle? c Explain why dividing the area of the rectangle by 2 gives us the area of the triangle. Practice 4 Estimate the area of each triangle to the nearest half centimetre.









79

7 Exploring measures

5 Circle the grey triangles that have an area of 9 cm2. You will need to measure the sides of the rectangles. a

c

b



d



e

6 Work out the length of the missing side.

m Area of this triangle is 12 m2



80

8m

7.1 Rectangles and triangles

Challenge 7 Draw some different right-angled triangles on the squares that have an area of 6 cm2.



How could you test that the triangle has an area of 6 cm2?



81

7 Exploring measures

8 Chata is painting triangular tiles to create a mosaic. This is the actual size of one tile. 2 cm 2 cm



One 15 ml pot of paint covers 18 cm2. 15 ml



What is the total number of paint pots that he will need to buy to cover 75 tiles?

9

2 cm

6 cm 4 cm

6 cm

a What is the area of the large square? b What is the area of each white triangle? c What is the area of the grey square?

82

7.2 Time

7.2 Time Exercise 7.2 Focus 1 Convert these times into minutes and seconds. a 2 minutes =

minutes and

b 2.5 minutes =

seconds

minutes and

seconds

c 3.25 minutes =

minutes and

seconds

d 3.75 minutes =

minutes and

seconds

2 Draw lines to match the same times. 2 hours and 45 minutes

1.25 days

1.25 hours

5.5 minutes

4 days and 12 hours

5 hours and 30 minutes 5.5 hours

1 hour and 15 minutes 2.75 minutes

4.5 days 2.75 hours

2 minutes and 45 seconds

5 minutes and 30 seconds 4 hours and 30 minutes

1 day and 6 hours 4.5 hours

83

7 Exploring measures

Practice 3 This line graph shows equivalent amounts of time in hours and in minutes. 65 60 55 50 45

Minutes

40 35 30 25 20 15 10 5 0



84

–5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Hours

Use the conversion graph to convert these times from hours to minutes. a 0.2 hours =

minutes

b 0.7 hours =

minutes

c 0.45 hours =

minutes

d 0.95 hours =

minutes

1

1.1

7.2 Time

4

2045 Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

31

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

31



This is a page from a calendar for the year 2045.



Use what you know about the months of a year and the number of days in each month to answer these questions. a Suggest two months that this calendar page could be for. Explain how you know. b If the calendar is from the second half of the year, what is the date circled? c Use what you know about years, months and days to work out how old each of these people will be on the circled date. i

Cheng was born on 11th July 2013, he will be and

ii

iii

years,

months

years,

months

days old.

Magda was born on 27th July 1975, she will be and

months

days old.

Halima was born on 2nd April 2006, she will be and

years,

days old.

85

7 Exploring measures

iv

Jack was born on 18th October 1972, he will be months and

v

years,

days old.

Stefan was born on 30th September 1966, he will be months and

days old.

Challenge 5 Six people are at a railway station waiting for six trains to different destinations.

Use the clues to work out the destination of each train. Destination

Departure time 11:48 12:18 12:58 13:23 13:53 14:28

Clues

86



The train for Barcelona leaves later than the train for Brussels, but before the train for Venice.



The train for Brussels leaves between 12 o’clock and 1 o’clock.



The train for Vienna leaves later than the train for Copenhagen, but before the train for Barcelona.



The train to Vienna leaves 40 minutes before the next train.



The train that leaves at 7 to 2 is going to a place with six letters in its name.



The train for Warsaw leaves before the train for Venice but later than the train for Barcelona.

years,

7.2 Time

Tip Use this logic table to work out which train left when. Put a cross in any places that cannot be the correct time for that train. Tick the time when you know it is correct for that train. 11:48

12:18

12:58

13:23

13:53

14:28

Barcelona Venice Brussels Vienna Warsaw Copenhagen 6 Make your own timetable. Use 24-hour clock times.

There are five stations on the line. You can name the stations.



There are three trains. They can go back and forth along the line as many times as you like.



These are the times between stations.

0.95 hours

1.2 hours

0.4 hours 1.15 hours



Trains must stop at each station for 2 minutes.



At the end of the line the train stops for 0.3 hours before going back.

87

7 Exploring measures



From

to

 :



From

to

 :

Tip Look at other bus or train timetables for ideas. In the first table, write the arrival and departure times from the first station to the fifth station. In the second table, write the arrival and departure times from the fifth station back to the first station.

88

8 Addition and subtraction (2) 8.1 Adding and subtracting decimal numbers Worked example 1

decimal place

Add together all the numbers greater than 0.7

trailing zero

0.507  0.8  0.38  0.09  0.747  0.699 0.8 + 0.747

Identify the numbers greater than 0.7

Estimate:

Estimate the answer and then add the numbers.

0.8 + 0.7 = 1.5 0.800 + 0.747 1.547 1

Align the decimal points correctly and add trailing zeros so all the numbers have the same number of decimal places. Compare your answer with the estimate to check it is reasonable.

89

8 Addition and subtraction (2)

Exercise 8.1 Focus 1 Abi pays $2 to buy some cheese that costs $1.35.

Complete this method to show how much change Abi gets. 2 1.35

+

–

0.9

+

+

+

+

+

0.05

=$

2 A chef is preparing school dinners.

She weighs 6.3 kg of pasta.



She needs 10 kg.



How much more pasta does she need?

3 Calculate. a 4.504 + 15.096 = b 4.985 – 2.347 = 4 Here are six number cards.



90

0.003

0.01

0.006

0.007

0.01

Use each card once to complete these two calculations. +

=

+

=

0.004

8.1 Adding and subtracting decimal numbers

Practice 5 A shop has these items for sale.



Coffee maker

$29.95

Toaster

$30.75

Can opener

$14.25

Ice cream maker

$26.80

Ravinder buys an ice cream maker and a coffee maker. a How much does he spend altogether? b How much change does he get from $60?

6 Ai cuts a piece of wood measuring 1 metre into three pieces. 0.2 m

?

not to scale



0.54 m

How long is the last piece?

91

8 Addition and subtraction (2)

7 Decimal pyramids are built like this. Each number is the total of the two numbers below.

2.9 1.2





1.7

Build these pyramids. b

a

1.07

4.8

5.9

7.323

5.6

6.12

3.6

0.023

Challenge 8 Find the missing digits. a

–

7

6 3

b

7

9

6

8

+

3

9 Find three decimals that add to 1.



92

6

3

Two of the decimals must have three decimal places and one must have two decimal places.

8

1

3

5

7

8.1 Adding and subtracting decimal numbers

10 Draw lines from two bags to each box to make the total mass written on the box.

2.9 kg

4.1 kg

0.34 kg

0.27 kg

1.19 kg

2.7 kg

3.8 kg

1.2 kg

4.9 kg

5.5 kg

8.7 kg

1.4 kg

4.8 kg

0.49 kg

5.99 kg

0.92 kg

0.86 kg



Label the last box with the total of the remaining two bags.



Use this space for working.



93

8 Addition and subtraction (2)

8.2 Adding and subtracting fractions common denominator Worked example 2 Calculate

5 2

–

denominator

4 3

Multiples of 2: 2, 4, 6 …

Find a common denominator by looking at multiples of 2 and 3.

Multiples of 3: 3, 6 … 5 2

=

15 6

4 3

=

8 6

5 2

–

4 3

=

=

Change

15 6

–

8 6

7 6

to equivalent fractions

=1

Simplify if possible. 1 6

1 Olaf draws a diagram to show

Write improper fractions as mixed numbers.

3 4



94

4 3

Subtract the numerators.

Focus



and

with a denominator of 6.

Exercise 8.2



5 2

What answer should Olaf write?

2 3

–  .

8.2 Adding and subtracting fractions

2 Calculate. a

1 4

4 5

b

+



1 4

2 3

+



c

3 8

+

2 3

c

4 5

–

2 3



3 Calculate. a

3 5

1 2

b

–



3 4

2 3

–





4 Write the letters of the calculations in the correct place in the diagram. A

2 3

3 8

B

+

3 8

2 5

+

Answer less than 1

C

2 3

+

5 15

D

3 4

Answer equal to 1

+

3 10

Answer more than 1

Practice 5 In this diagram the number in each box is the sum of the two missing numbers below it.

Write the missing numbers.

Tip 1 4

2 3

1 6

Write all the fractions in twelfths.



95

8 Addition and subtraction (2)

6 Chata solved this calculation.

3 5



What mistake has Chata made?



Correct Chata’s answer.

+

3 8

=

6 13

7 Alana spends

3 4

hour preparing an experiment and

doing the experiment.

3 10

hour

How long did Alana spend on the experiment altogether?

8 Each student in a club chooses to play cricket, tennis or rounders.

2 5 1 3

of the students play tennis. of the students play cricket.

What fraction of the students play rounders?

Challenge 9 Write the missing number.

4

10 Find the value of 1 –

96

3 8

–

2 5

7 1 – 10 = 20

8.2 Adding and subtracting fractions

11 Heidi drives for

5 3

hours, then takes a break. 7 4



She drives another



How long did Heidi spend driving?

hours to reach her destination.

12 Write the missing numbers to make this calculation correct.

5

+

1

7 = 10



Can you find more than one answer?



97

9

Probability

9.1 Describing and predicting likelihood Worked example 1 If you spun this spinner 20 times, how many times would you predict that it would land on ‘1’? 1

2

4

3

What is the probability of the spinner landing on ‘1’? 1 out of 4, or What is 5

1 4

1 4

or 25%.

of 20?

I predict that the spinner will land on ‘1’ five times.

equally likely outcomes   event mutually exclusive events   outcome probability   probability experiment

1 out of 4 equal sections of the spinner is ‘1’. There is a 1 out of 4 chance of the spinner landing on ‘1’. If there is a 1 out of 4 chance of the spinner landing on ‘1’ you can expect the spinner to land on ‘1’ for

of the spins.

Remember, this does not mean that the spinner will definitely land on ‘1’ five times. Each time 1 the spinner is spun there is a chance that it 4 will land on ‘1’, so it is random. The more trials you do the more likely it is that the spinner will 1 land on ‘1’ of the times. 4

98

1 4

9.1 Describing and predicting likelihood

Exercise 9.1 Focus 1 Arun, Marcus, Sofia and Zara have been investigating probability. They are taking shapes out of bags without looking. Draw a line from each child to a bag.

There is a 4 out of 4 chance of taking a pyramid from my bag.

A

B

There is a 2 out of 4 chance of taking a prism from my bag.

There is a 1 out of 4 chance that I shall take a cube out of my bag.

C

1

2

There is a 0 out of 4 chance of taking a pyramid from my bag.

D

3

5

4

99

9 Probability



One of these bags of shapes does not belong to any of the children. Complete the statements for this extra bag of shapes.



The probability of taking a prism from the bag is



 .



The probability of taking a 3D shape from the bag



is



The probability of taking a pyramid from the bag



is

 .

Tip Remember that cubes and cuboids are types of prism.

 .

2 Here is a set of cards used in a game.

8

3

2

2

4

2

3

5



Tasha is going to investigate the chance of taking a card with a triangle symbol.



Circle the cards with a triangle symbol.



Put an X next to the events below that could not occur at the same time as taking a card with a triangle symbol.



Taking a card with a square symbol



Taking a ‘2’ card



Taking a card with a value greater than 4



Taking a card with an odd number

3 Tear 10 small pieces of tissue paper.

Draw a circle on a piece of paper and place the paper on the floor.



Drop the tissue paper pieces onto the paper on the floor.



How many pieces landed inside the circle?



100

out of 10 pieces landed inside the circle.

9.1 Describing and predicting likelihood



Conduct a probability experiment to see how many pieces land inside the circle in each of 50 trials.



Record your results in this table. Number of pieces inside the circle

Tally

Total

0 1 2 3 4 5 6 7 8 9 10

Describe the results of your experiment.



The experiment showed that there was a 0 pieces of tissue landing in the circle.

out of 50 chance of



The experiment showed that there was a 10 pieces of tissue landing in the circle.

out of 50 chance of



The experiment showed



101

9 Probability

Practice 4 This is a net of a six-sided dice.

Number the dice so that:



There is a 50% chance of throwing a number greater than 5.



There is a greater than 50% chance that the number thrown will be even.



There is a 0% chance that the number thrown will be a multiple of 3.



The chance of throwing a number less than 0 is greater than 0%, but less than 50%.

5 Kapil says that rolling a ‘4’ on an ordinary 6-sided dice and rolling an odd number on a dice are mutually exclusive.



Is Kapil correct?



Complete the sentence.



Two events are mutually exclusive when they

6 1

2 3

5 4



What is the chance of spinning a 2?



If you spin the spinner 50 times how many 2s would you expect to see?



Use a pencil and paperclip like this to complete the spinner.

/



1

2 3

5

102

4

9.1 Describing and predicting likelihood



Conduct the experiment by using the spinner 50 times. Use this space to draw a tally chart for your outcomes.



Describe the results of your experiment.



103

9 Probability

Challenge 7 Some children are members of a tennis club. This Carroll diagram shows the children in the club. Under 13 years old

Not under 13 years old

Boys

Cheng

Farrukh

Girls

Gemma

Dan

Sunita

Scott

David

Talia

a How many children are members of the club? b One child is chosen at random for a lesson. What is the chance that: i

The child is a girl under 13 years old?

ii

The child is a boy not under 13 years old?

iii

The child is a boy?

iv

The child is not under 13 years old?

v

The child is not a boy under 13 years old?

8 Lola has six t-shirts. She takes a t-shirt at random.



104

9.1 Describing and predicting likelihood



These are four events that could occur when Lola takes a t-shirt.



Event A: The shirt has a picture of an animal on it.

Tip



Event B: The shirt has a picture of fruit on it.



Event C: The shirt is striped.



Event D: The shirt has a collar.



Tick the pairs of events that are mutually exclusive for Lola’s set of t-shirts.

If Event A and Event B are mutually exclusive put a tick (✓) in the grey box.

A B C D E



Write your own Event E which is mutually exclusive to Events A and B, but not mutually exclusive to Events C and D.



Event E:



Complete the table for your own event.

9 Marco has put 4 different coloured balls into a bag.

He is going to conduct a chance experiment and record the outcomes when he takes and replaces a ball 100 times from the bag.



These are his predictions: • I expect to take a red ball 20 times. • I expect that none of the balls I take will be blue. • I expect to take a yellow ball 25 times. • I expect to take a purple ball 50 times. • I expect to take a green ball 5 times.

105

9 Probability

a Colour the picture to show what colour balls Marco put into the bag. b Add 5 more coloured balls to the bag. Write your predictions for how many of each colour you expect to take if you repeated Marco’s experiment now.

I expect to take

red balls.



I expect to take

blue balls.



I expect to take

yellow balls.



I expect to take

purple balls.



I expect to take

green balls.

c If you carried out the experiment would you expect your results to be exactly as you predicted? Why?

106

10 Multiplication and division (1) 10.1 Multiplication Worked example 1

product

Calculate 256 × 27 Estimate 250 × 20 = 5000 and 250 × 30 = 7500

Start by estimating the size of the answer.

The answer is between 5000 and 7500. 2

5

6

×

2

7

1

7

9

2

5

1

2

0

6

9

1

2

• Multiply 256 by 7 • Multiply 256 by 20 • Add the two answers together The order you do the multiplications does not matter.

1

Answer: 6912

Use your estimate to check that your answer is reasonable.

107

10 Multiplication and division (1)

Exercise 10.1 Focus 1 Complete the cross number puzzle. 1

2

3

4 5

6 7 9

8

10

11

12 13

14

15

ACROSS

DOWN

1.

171 × 9

1.

361 × 4

3.

8 × 9

2.

158 × 6

4.

528 × 8

3.

927 × 8

5.

502 × 9

6.

748 × 2

7.

253 × 5

7.

3×6

9.

732 × 4

8.

628 × 8

11. 224 × 7

9.

513 × 5

12. 128 × 4

10. 956 × 3

13. 157 × 4

12. 117 × 5

14. 774 × 2 15. 6 × 9

108

10.1 Multiplication

Practice 2 Calculate each product.

You must write an estimate before doing the calculation. a 1546 × 7 = b 2398 × 8 = c 3594 × 6 =

3 Write in the missing digits to make this calculation correct. 7 6 × 1



0

3

2

4 Nailah estimates 2999 × 70 = 210 000

Has she made a good estimate?



Explain your answer.

5 Use the digits 0, 1, 5 and 9 to complete this calculation. ×



= 1350

6 Hassan finds the product of two multiples of 10.

The answer is 12 000.



List all the calculations that give his answer.



109

10 Multiplication and division (1)

Challenge 7 Ella and Roz complete the same multiplication. Ella 7

Roz 3

×

4

5

2

4

× 1

2

9

3

8

0

1

4

6

9

0

0

1

7

6

2

8

0

1

1



Who has the correct answer?



What mistake has the other girl made?

7

1

3

4

5

2

4

4

6

9

0

0

2

9

2

8

0

7

6

1

8

0

1

1

8 Find the product of 6589 × 37 9 The distance from London to Budapest is 1723 kilometres.

Mary lives in Budapest and travels to London and back six times. How far does she travel?

10 The table shows ticket sales during one day at four theatres. Theatre

110

Apollo

Lif

Legend

Mani

Number of tickets sold

2108

1935

2245

1649

Cost of ticket

$45

$39

$42

$47

10.2 Division



Which theatre took the most money during the day?



You must show all your working.



10.2 Division Worked example 2

dividend  divisor

A coach operator puts a first aid kit on 378 coaches.

quotient  remainder

The first aid kits are sold in boxes of 18. How many boxes are needed? Estimate: 380 ÷ 20 = 19 so the answer will be approximately 19 21 18 378 – 360 18 –18

18 × 20 18 × 1

0

Start by making an estimate. There are twenty 18s in 378. Record 2 tens on the answer line. Subtract 360 (18 × 20) from 378 to leave 18. There is one 18 in 18. Record 1 one on the answer line. Subtract 18 (18 × 1) from 18 to leave 0. There is no remainder.

Answer: 21 boxes are needed

Exercise 10.2 Focus 1 Calculate 837 ÷ 9

111

10 Multiplication and division (1)

2 Four friends have lunch each day at a café. The total cost for the week is $152. They share the cost equally.

How much does each person pay?

3 Henryk pays $9 each week to go to the gym. He has paid $747 so far.

How many weeks has he been to the gym?

4 A team of volunteers made a total of $992 by selling T-shirts for charity.



Each T-shirt costs $8.



How many T-shirts did the volunteers sell?

Practice 5 Work out the value of the missing digits. a 49

÷ 5 = 98 remainder 3

b 65

÷ 9 = 72 remainder 6

6 Calculate 936 ÷ 12

112

10.2 Division

7 An art gallery collected $600 in entrance fees in 1 hour.

Art Gallery

entrance fee $12 per person



How many people paid to enter the gallery during the hour?

8 A teacher needs 240 exercise books.

The exercise books are sold in packets of 16.



How many packs must the teacher buy?

Challenge 9 Leanne and Carrie complete the same division. Leanne

720 ÷ 24 720 ÷ 4 ÷ 6



Carrie

720 ÷ 24 720 ÷ 20 ÷ 4

= 180 ÷ 6

= 36 ÷ 4

= 30

=9



Who has the correct answer?



What mistake has the other child made?



113

10 Multiplication and division (1)

10 Find the missing digits. 2 4



3 4

r2 2

11 Use the digits 1 to 6 to complete these divisions. You must use each digit once. a 59 b c 47

÷4=

48

89 ÷ 5 = 77 r ÷3=1

8 r2

10.3 Tests of divisibility Worked example 3

divisible  

Here is a Venn diagram for sorting numbers.

factor multiple

Write each number in the correct place on the diagram.

test of divisibility

3741

Venn diagram

588

1679

divisible by 3 divisible by 6

114

1569

1092

10.3 Tests of divisibility

Continued

divisible by 3

Find the numbers that are divisible by 3 by finding the sum of the digits, for example:

divisible by 6

3 + 7 + 4 + 1 = 15 so 3741 is divisible by 3.

588

1569

1092

588, 1092 and 1569 are also divisible by 3. 588 and 1092 are also divisible by 6 because they are even numbers.

3741 1679

1679 is not divisible by 3.

Exercise 10.3 Focus 1 Which of these numbers are divisible by 3?

Explain how you know.



4563

54 689

234 567

2 Write a digit in each box so that all the numbers are divisible by 3. Can you find more than one answer? a 7

b 501

23

7

3 Here are four labels.

even

divisible by 9

not divisible by 9

not even

115

10 Multiplication and division (1)



Write each label in the correct place on the Carroll diagram.

2322

2348 321 426 770 679

 2331

723 142 4867 126 147

Practice 4 Here are five numbers.

64



Tick the statement that describes all the numbers.

128

240

352

424

A They are all divisible by 6 B They are all divisible by 7 C They are all divisible by 8 D They are all divisible by 9 5 Tick in the correct cells to show whether these numbers are divisible by 3, 6 and 9. 3 21 471 482 211 152 214

116

6

9

10.3 Tests of divisibility

6 Here is a Venn diagram.

Put these numbers on the diagram.



159

204

146

324

222

divisible by 6

189

divisible by 9

divisible by 3

Challenge 7 Kojo says, ‘Multiples of 6 can never end in a 3.’ Is he right?

Give a reason for your answer.

8 Here are four digits.

1



Use these digits to make three 3-digit numbers. You can use each digit more than once in any number, but you must use all the digits at least once and all three numbers must be different.

2

5

7

a number divisible by 3 a number divisible by 6



a number divisible by 9

117

10 Multiplication and division (1)

9 Here is a Carroll diagram with four sections A, B, C and D. divisible by 9

not divisible by 9

divisible by 3

A

B

not divisible by 3

C

D

a Write a 5-digit number in sections A, B and D. b You cannot write a number in Section C. Explain why not.

118

11

3D shapes

11.1 Shapes and nets Worked example 1

compound shape prism  surface area

Describe this compound shape.

This compound shape is made from a cone and a cylinder.

Think how you can split the compound shape into simpler 3D shapes that you know.

119

11 3D shapes

Exercise 11.1 Focus 1 Complete these descriptions of compound shapes. a



This compound shape is made from a

and a

 .

b



This compound shape is made from two

 .

c



This compound shape is made from a

and a

 .

2 The diagram shows four shapes A, B, C and D.



120

It also shows four sketches of nets i, ii, iii and iv. A

B

C

D

i

ii

iii

iv

Draw a line to match each shape to the correct net.

11.1 Shapes and nets

3 Write down the smallest number of unit cubes that must be added to these shapes to make cuboids. a

 

c



b

 

 

Practice 4 Sketch a compound shape that is made from these simple shapes. a three different cuboids



121

11 3D shapes

b two identical cones

c two identical pyramids.

5 Describe and sketch a net of these shapes. a triangular prism



122

11.1 Shapes and nets

b hexagonal prism.

6 Describe how could you work out the surface area of the shapes in question 5. 7 Write down the smallest number of unit cubes that must be added to these shapes to make cuboids. a

 

c

b

 

 

123

11 3D shapes

Challenge 8 Sketch a net for these shapes. a

b



124

11.1 Shapes and nets

9 Write down the smallest number of unit cubes that must be added to these shapes to make cubes. a

 

c

b

 

 

125

11 3D shapes

10 These two shapes are made from unit cubes.



Fran takes the shapes apart and uses all the unit cubes to make a cuboid.



Draw a sketch to show two different cuboids she can make with all the cubes. Use isometric paper if you have some.



126

11.2 Capacity and volume

11.2 Capacity and volume Worked example 2

capacity  volume

The diagram shows some water in a jug. ml

500 400 300 200 100 0

a What is the capacity of the jug? b What is the volume of water in the jug? a 500 ml

500 ml is the maximum the jug can hold

b 300 ml

The scale shows the water is at the 300 ml mark.

Exercise 11.2 Focus 1 For each of these jugs write down i

the capacity of the jug

ii the volume of water in the jug. a

ml

500 400 300 200 100

b

ml

100 90 80 70 60 50 40 30 20 10

i





ii





c

ml

5000 4500 4000 3500 3000 2500 2000 1500 1000 500

127

11 3D shapes

2 Write the capacity of the container and the volume of the liquid for each of these diagrams. A



litres

Capacity

B   



200 180 160 140 120 100 80 60 40 20 Capacity

C   



litre

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

ml

1600 1400 1200 1000 800 600 400 200

Capacity

E   

Volume

ml

Capacity

D   



2 1.75 1.5 1.25 1 0.75 0.5 0.25

litres

Volume

Volume

Volume

1.2 1 0.8 0.6 0.4



128

0.2

Capacity

Volume

11.2 Capacity and volume

3 Show how these bottles should be organised and grouped in the Venn diagram. Bottle B

Bottle A

Bottle C

Bottle D 0.09 litres

300 ml 200 ml

75 ml 400 ml

750 ml

500 ml

1 litres

Bottle E

Bottle F

Bottle G

Bottle H

80 ml 0.2 litres 0.5 litres 50 ml



300 ml

80 ml

Capacity less than 500 ml

0.2 litres

0.75 litres

Volume less than 100 ml



Draw one more bottle in each of the four sections of the Venn diagram. Label the bottles with their capacity and the volume of liquid inside.

129

11 3D shapes

4 Complete this table.

Remember that 1000 ml = 1 litre. millilitres

litres and millilitres

litres

4100 ml

4 l 100 ml

4.1 l

1500 ml

1 l   ml 3 l 600 ml 2.5 l

400 ml

0 l 400 ml

9600 ml 5 What number is the arrow pointing to on each of these scales? Look at the tip boxes for help. a

b

200 ml 

Tip

100 ml

There are four spaces for an increase of 100 ml, so each increment is worth 100 ÷ 4 = 25 ml.

400 ml 

300 ml

c

3 l    

2l

130

Tip There are five spaces for an increase of 100 ml, so each increment is worth 100 ÷ 5 = 20 ml. Tip There are five spaces for an increase of 1 l, so each increment is worth 1 ÷ 5 = 0.2 l.

11.2 Capacity and volume

Practice

ml

6 Zara and Sofia are looking at this question.

What is the volume of water in this jug?



Read what they say.

100 90 80 70 60 50 40 30 20 10

I think the volume of water is 79 ml.





I think the volume of water is 78 ml.

a Who is correct? Explain why. b Explain the mistake that the other person has made. c What volume of water must be added to the jug to fill it to capacity?

131

11 3D shapes

7 For each of these jugs write down: i

the capacity of the jug

ii the volume of water in the jug. a

ml



b



200

litres

c

4

800

ml

600

3

100

400 2

200

1

i





ii





8 What volume of water must be added to the jugs in question 7 to fill them to capacity? a

b

c

9 a What is the total capacity of the 5 cans of oil below? A 2.5 litres

B 2.5 litres

C

D

2.5 litres

litres E

2.5 litres

2.5 litres

b Estimate the volume of oil needed to fill each can. Give your answers in millilitres. a

132

b

c

d

e

11.2 Capacity and volume

Challenge 10 Dakarai needs 3 litres of water.

He only has the water shown in the measuring jugs. 3

litres litres

1.8 2 1.2 1

0.6



            Does he have enough water? Explain your answer.

11 Rhian has a bucket with a capacity of 10 litres. The bucket is

3 5

full of water. 1 3



Wyn has a bucket which is



He has the same volume of water in his bucket as Rhian.



What is the capacity of Wyn’s bucket?

full of water.

12 Elin has these four measuring cups, A, B, C and D. The capacity of each cup, in millilitres, is shown.



240 ml

160 ml

120 ml

60 ml

A

B

C

D

133

11 3D shapes



Explain how Elin can use the cups to accurately measure out these volumes: a 40 ml b 180 ml c

80 ml

d 100 ml e 20 ml 13 Jenny was asked this question:

Is the capacity of a container always greater than the volume of the liquid inside?



Jenny wrote: The capacity of a container must always be greater than



Think carefully about capacity, volume and Jenny’s answer. Write an improved answer of your own.



134

the volume of the liquid inside because the capacity is the maximum the container can hold.

11.2 Capacity and volume

14 Kabir has three fuel containers. One has a capacity of 7 litres, one has a capacity of 4 litres and one has a capacity of 3 litres. The 7 litre container is full, the other two containers are empty. None of the containers have a measurement scale.

How can Kabir transfer the fuel so that two of the containers contain a volume of 2 litres each, and the other contains a volume of 3 litres, without having to estimate?



135

12 Ratio and ­proportion 12.1 Ratio equivalent ratio  ratio Worked example 1

simplest form

Write the ratio 6 : 18 in its simplest form. ÷6

6 : 18

÷6

6 is a factor of both 6 and 18

1:3

To find an equivalent ratio, we divide both quantities in the ratio by the same number. Answer 1 : 3

Exercise 12.1 Focus 1 Look at these shapes.



Write in its simplest form: a the ratio of circles to squares b the ratio of squares to pentagons c the ratio of pentagons to squares.

136

12.1 Ratio

2 Write these ratios in their simplest form. a 3 : 15

b 21 : 49

c 24 : 18

d 24 : 6

3 Write the missing numbers. a 2 : 3 = 4 :

b 5 : 7 =

: 14

c 3 : 5 = 12 :

Practice 4 Look at this recipe for pasta sauce.

Pasta sauce 300 g tomatoes 120 g onions 75 g mushrooms

a Write the ratio of tomatoes to mushrooms in its simplest form. b Write the ratio of mushrooms to tomatoes in its simplest form. 5 Jamie mixes 2 parts of red paint with 3 parts of blue paint to make purple paint.

He uses 12 cans of blue paint.



How many cans of red paint does he use?



137

12 Ratio and proportion

6 There are 2 milk chocolates for every 3 dark chocolates in a box of chocolates.

There are 8 milk chocolates in the box.



How many chocolates are in the box altogether?

7 On a school visit, there is 1 teacher for every 8 students.

There are 96 students on the visit.



How many teachers are there on the visit?

8 Keon prepares a picnic.



Each person will get:

3 sandwiches



2 cartons of orange juice



1 banana

Ollie packs 45 sandwiches. a How many cartons of orange juice does he pack? b How many bananas does he pack?

Challenge 9 Jodi makes a fruit drink using oranges and lemons.

For every 1 lemon she uses 4 oranges.



She uses 20 pieces of fruit altogether.



How many oranges does she use?



138

12.1 Ratio

10 Here is a pattern and some statements.



One statement does not describe the pattern. A Ratio of triangles to circles is 1: 4

B 1 in every 5 shapes is a triangle

C 20% of the pattern is triangles

D 1 out of 5 shapes is a triangle

E Ratio of triangles to circles is 4 :16

F

1 5

G 2 in every 8 shapes are triangles

H

8 10

I

J

4 out of 20 shapes is a triangle

80% of the pattern is circles

of the pattern is triangles of the pattern is circles

a Which statement does not describe the pattern? b Correct this statement.



11 Two numbers are in the ratio 3 : 7.

One of the numbers is 42.



Find two possible values for the other number.



and

12 The sides of an isosceles triangle are in the ratio 2 : 2 : 1.

2

2

Not drawn to scale

1



The shortest side is 5 cm long.



What is the perimeter of the triangle?

cm

139

12 Ratio and proportion

12.2 Direct proportion Worked example 2

direct proportion

This recipe gives the quantities to make 18 doughnuts.

Doughnuts [makes 18 doughnuts] 300 g flour 60 g margarine 90 ml milk 75 g sugar 3 eggs

What quantities should Imran use to make 6 doughnuts? 6 is one-third of 18 so Imran must divide all the quantities by 3:

140

100 g flour

300 ÷ 3 = 100

20 g margarine

60 ÷ 3 = 20

30 ml milk

90 ÷ 3 = 30

25 g sugar

75 ÷ 3 = 25

1 egg

3÷3=1

enlarge  proportion

12.2 Direct proportion

Exercise 12.2 Focus 1 Hassan goes shopping at a shop where each item costs $5. Complete the table. Number of items

1

2

3

4

Cost in $ 2 A bottle-making machine makes 60 bottles in 5 hours.

a How many bottles will it make in 20 hours? b How long will it take to make 30 bottles? 3 Here are two squares.

Not drawn to scale



3 cm



The ratio of the side length of the small square to the side length of the large square is 1 : 3.



What is the side length of the large square?



141

12 Ratio and proportion

Practice 4 A picture of a cat is one-fifth of the size of the real cat. a The cat’s body is 9 cm long in the picture. How long is the body of the real cat? b The real cat’s tail is 30 cm long. What is the length of the tail in the picture? 5 This is a recipe for pancakes for 8 people.

Pancakes [for 8 people] 100 g flour 2 eggs 300 ml milk



142

What quantities should Mira use to make pancakes for 4 people?

12.2 Direct proportion

6 Kojo draws a rectangle 1 cm by 2 cm.

He enlarges the rectangle to make three more rectangles.



Which of these rectangles can he draw? Explain your answer.



A: 2 cm by 4 cm

B: 5 cm by 20 cm



C: 9 cm by 18 cm

D: 4 cm by 8 cm

7 Here is a recipe for raspberry ice cream.

Receipe

Receipe

Raspberry ice cream Serves 8 people 1 2

litre cream

1 kg raspberries

250 g sugar

a Hong makes enough ice cream for 4 people.

How many grams of raspberries does she use?

b Hassan also makes raspberry ice cream. He uses 1kg sugar. What mass of raspberries does he use?

143

12 Ratio and proportion

Challenge 8 Here is rectangle A. 6 cm A



2 cm

Not drawn to scale



Rectangle A is enlarged to make rectangles B and C.



Complete the table. Ratio of lengths

Width in cm

A

B

C

Length in cm

2

A to B = 1 : 5

A to C = 1 : 10

9 Here is a recipe for chocolate ice cream.

Receipe Chocolate ice cream Serves 8 people 420 ml cream 400 ml milk 4 egg yolks 120 g chocolate 100 g sugar



144

Receipe

Perimeter in cm

12.2 Direct proportion

a Pierre has only 40 g of chocolate to make his ice cream.

How much cream should he use?

b Maxine makes chocolate ice cream for 12 people.

How much milk does she use?

10 Saif makes milkshakes using 1 part syrup to 7 parts milk. a How much syrup should he add to these glasses of milk? A

700 ml milk



ml milk

B

C

630 ml milk

350 ml milk

ml milk

ml milk

D

490 ml milk ml milk

b What proportion of each milkshake is syrup?

145

13

Angles

13.1 Measuring and drawing angles Worked example 1

protractor

a Measure angle x.

X°

b Draw an angle of 135° a 90 100 110 70 12 270 260 2 0 0 8 60 2 50 13 90 2 24 0 0 50 0 02 3 0 30 1 3

20

03

30

33

20

40

0

0

23

0

260 270 280 29 0 250 30 0 0 00 90 80 7 0 31 10 1 60 0 50

0 33

40

13

0

0 32

22

40

0

01

15

350 340 10 20 30

170 1 60

12

24

01

146

x = 60°

170 180 190 20 0 2 1

90 00 1



160

02

X°

350 3

0

21

10

0

15

22

0

14

40

80

Place your protractor over angle x so that the centre of your protractor is at the point of your angle. Make sure the horizontal arm of your angle is lined up exactly with 0°. As the angle opens clockwise, use the numbers on the outside circle.

13.1 Measuring and drawing angles

Continued b

90 100 110 12 0 80 270 260 25 60 2 0 13 02 9 02 40 0 50 30 23 0 0 31

20

40 40

260 270 280 29 0 250 3 00 0 90 80 24 7 0 100 0 31 1 60 0 50 0

0 32

Start by drawing a horizontal line. Place the centre of your protractor on the right end of the line, with the left end of the line at 0°. Measure 135° clockwise, using the outside numbers. Mark a small line as this point.

0

10

350 3

0

350 340 10 20 30

13

0 33

170 180 190 20 0 160 2 10 0 170 90 15 1 6 22 00 1 01 0 02 50 0 14 21 14 0 0 22

30 4 33 03 0 20

80

23

70

Now take away the protractor and join your mark to the right end of the horizontal line. Draw an angle arc and write 135° onto you diagram.

12

01

135°

Exercise 13.1 Focus 1 Measure the size of each of these acute angles.



Circle the correct answer for each one.

a°

      a = 20° a = 30°

b°

    

 

b = 55° b = 65°

147

13 Angles

c°



c = 70° c = 78°

2 Measure the size of each of these obtuse angles.

Circle the correct answer for each one.

d°







148

  

d = 100° d = 110° e°

  

e = 160° e = 170°

f°

f = 132° f = 142°

13.1 Measuring and drawing angles

3 Measure the size of each of these reflex angles.

Circle the correct answer for each one. g°



    g = 210° g = 220° h°



    h = 260° h = 270° i°



i = 340° i = 330°

149

13 Angles

Practice 4 Measure the size of each of these angles.

x°



  



y°



z°



150

13.1 Measuring and drawing angles

5 Draw angles of the following sizes. a 10°

b 165°

c 230°

d 330°

6 a Measure the angles x and y in this diagram.

x°

y°

b Which calculation can you do to check that your answers to part a are correct?

151

13 Angles

Challenge 7 The diagram shows a ski jump ramp.

x

y



In a competition, the angle marked x must be between 36° and 38°, and the angle marked y must be between 7° and 12°.



Can this ramp be used in a competition? Explain your answer.

8 a Measure the angles a, b and c in this diagram.

a c



b

a=

b=

c=

b Explain why the angles should add up to 360°. Use this fact to check your accuracy.

152

13.1 Measuring and drawing angles

c Show that angle b is three times the size of angle a. d Write down the missing numbers in these statements: i angle c is

times the size of angle b.

ii angle c is

times the size of angle a.

9 The diagram shows two triangles. j g

d a



Triangle 1 

e

k

b

c

h

i f

  Triangle 2 

l

a Complete these tables which show the sizes of all the angles. Also fill in the totals of the angles shown.



Triangle 1 a=

d=

a+d=

b=

e=

b+e=

c=

f=

c+f=

a+b+c=

d+e+f=

a+b+c+d+e+f=

g=

j=

g+j=

h=

k=

h+k=

i=

l=

i+l=

g+h+i=

j+k+l=

g+h+i+j+k+l=

Triangle 2

153

13 Angles

b What do you notice about the totals that you have found in both tables? c Do you think that these totals will be the same for any triangle that you draw? Explain your answer.

13.2 Angles in a triangle Worked example 2

equilateral triangle

Work out angle y in this triangle.

scalene triangle

43°

y

154

isosceles triangle

75°

43 + 75 = 118°

Add together the two angles that you know.

180 – 118 = 62°

The angles in a triangle add to 180°, so subtract the total so far from 180°.

y = 62°

Write down the value of y.

13.2 Angles in a triangle

Exercise 13.2 Focus 1 Complete the workings to find angle x in each of these triangles. a



80°

b

x

40°

x

65°



80 + 40 = c

65 + 45 =



180 –

180 –



x=

=

 °

45°

 °

=

 °

 ° x=  °

2 Complete the workings to find angle y in each of these triangles. a

y

b

35°

30°



90 + 30 =



180 –



y=

y

 °

=

 °

 °

90 + 35 =

 °

180 –

y =

=

 °

 °

155

13 Angles

3 Complete the workings to find angle z in these triangles. a

b



z

80°

30° z



30 × 2 =



180 –



z=

 °

=

180 –

=

 °

 °

 °

÷2=

y =

 °

 °

Practice 4 Work out the lettered angle in each of these triangles. a



m

b

120° n 35°

48° 56°



m=

n=

5 This is part of Juan’s homework.

156



Question This triangle is isosceles.





Work out angles a and b.



Solution

a = 62°





62 × 2 = 124





180 – 124 = 56





b = 56°

a

b

62°

13.2 Angles in a triangle

a Explain the mistake that Juan has made. b Write down the correct values of a and b.

a =

b =

6 Helmut builds a swing in his garden. The diagram shows the largest angle that his swing can turn through.

115°

m



Helmut finds out that the swing is safe when the angle marked m is greater than 30°.



Is Helmut’s swing safe? Explain your answer.



157

13 Angles

Challenge 7 The diagram shows a right-angled triangle. y x



Zara is investigating different values for x and y. a Complete the table for these different values for x and y. x

40°

y

50°

30°

55°

Tip

24° 72°

61°

When x = 40°, y = 180 – 90 – 40 = 50°

b

In a right-angled triangle, the sum of x and y is always 90°.





158

Show that Zara is correct using the values in the table.

13.2 Angles in a triangle

c

Explain why the sum of x and y is always 90°. d Is it possible for x or y to be greater than 90°? Explain your answer.

8 Is it possible to draw a triangle with angles of 48°, 72° and 50°? Explain your answer. 9 In a right-angled triangle, the angles are d, e and f. Work out the values of d, e and f when: a d is the largest angle and e and f are the same size. b f is the smallest angle, e is two times the size of f and d is three times the size of f.

159

14 Multiplication and division (2) 14.1 Multiplying and dividing fractions denominator  numerator  operator

Worked example 1 Calculate

4 5

proper fraction   unit fraction

×4

Use a diagram in your answer. 4 5

×4=

=3

16 5

Multiply the numerator by the whole number. Change the improper fraction to a mixed number.

1 5 4 5 4 5 4 5 4 5

160

You can use other types of diagram, including a number line. = 16 5

This shows + 45 0

4 5

× 4 as repeated addition of

+ 45 4 5

+ 45 8 5

+ 45 12 5

16 5

4 5

14.1 Multiplying and dividing fractions

Exercise 14.1 Focus 1 Write an addition sentence and a multiplication sentence for the shaded parts in this diagram.

2 Calculate. a

b 2 3

3 5

÷3



÷4



3 Find the missing numbers. a

1 5

of 20 =

b

3 4

× 20 =

4 Write a division sentence for this diagram.

Practice 5 Calculate. a

7 9

÷3

b

3 7

÷4



161

14 Multiplication and division (2)

6 My family eats

3 4

of a box of cereal each week.

How much cereal do they eat in 4 weeks?

7 a Complete the multiplication table. 1 8

×

3 8

5 8

3 4 5 b Which two calculations give the same answer? 8 Leila cuts a

3 4

metre length of ribbon into 5 equal pieces.



What is the length of each piece of ribbon?



Give your answer as a fraction of a metre.

Challenge

3 8

9 Omar says, ‘  multiplied by 4 equals

Is Omar correct?



Draw a diagram to explain your answer.



162

12  .’ 32

7 8

14.1 Multiplying and dividing fractions

10 Write the letter of each expression in the correct cell in the table below. A

2 3

÷8

B

1 3

÷2

C

3 4

D

5 6

÷ 10

E

2 3

÷2

F

2 3

Answer

1 6

Answer

1 3

Answer

÷9 ÷4

1 12

11 Here are six numbers.

14



Use each of these numbers once to make these statements correct.

16

18

21

a

2 3

of

=

c

2 5

of

=

24

40

b

3 4

of

=

12 Use the digits 1, 2, 3 and 4 to complete this calculation.

÷

=

6



163

14 Multiplication and division (2)

14.2 Multiplying decimals Worked example 2

decimal  decimal place

Mia is thinking of a number.

decimal point  product

She says, ‘If I divide my number by 5 the answer is 73.45’ What number is Mia thinking of? ÷5

Multiplication and division are inverse operations, so you need to calculate 73.45 × 5.

73.45 ×5

73.45 × 5 = ? Estimate:

Start with an estimate.

70 × 5 = 350 and 80 × 5 = 400 The answer is between 350 and 400.

You can round 73.45 down to 70 and up to 80, then multiply both numbers by 5. The correct answer will be between these two values.

7 × 3

3

5

5

5 6

7

2

1

2

2

Answer: 367.25

164

4

Check the answer is between the two estimated values.

14.2 Multiplying decimals

Exercise 14.2 Focus 1 Calculate. a 24.1 × 6

b 18.2 × 4

c 40.9 × 5







2 One bracelet costs $4.65. How much do five bracelets cost? 3 Heidi helps in a shop for 8 hours. She is paid $7.55 for each hour she works.

How much does she earn?

4 Draw a line from each calculation to the correct label for the answer.

12.45 × 9

answer less than 100



15.4 × 6

answer equal to 100



13.84 × 7

answer more than 100



12.5 × 8

Practice 5 Write the three missing numbers in this multiplication grid. ×

6

0.56

3.36

0.27

0.69

4

3.92

1.08

4.14

7

1.89

2.76



165

14 Multiplication and division (2)

6 Which calculation is the odd one out? Explain why.

41.5 × 5

32.7 × 4

16.25 × 8

13.35 × 6

14.2 × 9

7 Calculate. a 46.3 × 11

b 39.3 × 23

c 23.8 × 35







8 Trudy buys 4.25 kg fish to make fish pies.

If the fish costs $13 per kilogram, how much does she pay?

Challenge 9 Pedro sells rugs for $24.75 each.

In one week he sells 26 rugs.



How much money does he make in that week?

10 Salma is thinking of a number.

She says, ‘If I divide my number by 16 the answer is 15.54.’



What number is Salma thinking of?

11 Hassan has 18 crates each with a mass of 0.3 tonnes.

The maximum load on his lorry is 5 tonnes.



Can Hassan load the crates safely? Show your working to explain your answer.



166

14.3 Dividing decimals

12 A school chef has 75 kg of flour.

She uses an average of 5.35 kg flour each week.



There are 13 weeks in the school term.



Is she likely to have enough flour for the term?



Show calculations to explain your answer.



14.3 Dividing decimals Worked example 3 Two different shops sell modelling clay.

dividend divisor quotient

Shop A sells 4 packets of clay for $10.28 Shop B sells 3 packets of clay for $7.80 Which shop has the better deal? Explain your answer. 1 packet in shop A costs: 2.5 7 2 2 4 1 0.2 8

1 packet in shop B costs: 2.6 0

To compare the prices, work out how much 1 packet costs in each shop. Divide the total cost by the number of packets.

1

3 7.8 0

Answer: The cost of 1 packet in shop A is cheaper, but you must buy 4 packets at a time.

Remember you will also need to think about how many packets you need.

If you only need 3 packets you would go to shop B.

167

14 Multiplication and division (2)

Exercise 14.3 Focus 1 Calculate. a 14.4 ÷ 3

b 22.4 ÷ 7

c 34.4 ÷ 8







2 Work out the answer to each question to help you find your way through the maze. start

12.4 ÷ 4

3.1

4.1

30.6 ÷ 9

5.4

5.7

3.6

3.4

31.8 ÷ 6

17.1 ÷ 3

44.8 ÷ 8

5.5

5.6

11.2

5.2

17.4 ÷ 6

12.5 ÷ 5

36.4 ÷ 7

5.2

3.4

end

3 Nine notebooks cost $14.58.

If each notebook costs the same, what is the price of one notebook?

4 A regular pentagon is marked out on the playground.

The perimeter of the pentagon is 24.95 metres.



What is the length of one side of the pentagon?



168

14.3 Dividing decimals

Practice 5 Work out the answer to each calculation and write it in the correct part of the table.

76.32 ÷ 8 =

56.2 ÷ 5 =



24.15 ÷ 7 =

61.2 ÷ 3 =

Answer less than 10

Answer between 10 and 20

Answer more than 20

6 Pierre is thinking of a number.

He multiplies his number by 9 and his answer is 147.6.



What number is Pierre thinking of?

7 Find the odd one out

15.6 ÷ 6    16.8 ÷ 7    20.8 ÷ 8    23.4 ÷ 9



Explain your answer.

8 Work out the answers to these calculations.

Five answers are on the grid. Which answer is missing? 6.12

6.04

6.15

6.23

6.07

6.17

a 91.05 ÷ 15

b 73.44 ÷ 12

c 87.22 ÷ 14







d 78.52 ÷ 13

e 111.24 ÷ 18

f







98.72 ÷ 16

169

14 Multiplication and division (2)

Challenge 9 A chef needs lots of bags of rice.

Deal A: buy 4 bags for $5.08



Deal B: buy 5 bags for $6.25



Which deal should he choose?



Explain your answer by showing your calculations.

10 Find the missing digit. 3 •6 7 2



•2

11 Look at these four calculations.

One of them is wrong.



281.7 ÷ 3 = 93.9

359.1 ÷ 9 = 39.9



939.7 ÷ 9 = 93.3

117.9 ÷ 3 = 39.3



Identify the incorrect calculation without working out the answers.



Explain your answer.

12 A dress maker cuts pieces of ribbon 15 centimetres long from a roll of ribbon that is 5.625 metres long.

170

What is the greatest number of 15-centimetre pieces she can cut from the roll of ribbon?

Tip Change metres to centimetres.

15

Data

15.1 Bar charts, dot plots, waffle diagrams and pie charts Worked example 1 How many of the people represented by this pie chart chose ‘mountains’ as their favourite place?

bar chart  data  dot plot pie chart   waffle diagram

forest city ocean mountains

Pie chart showing the favourite places of a group of 20 people

A quarter of the pie chart is labelled 1 4

‘mountains’, so of the people chose the mountains as their favourite place. 1 4

= 25%. 25% of the people chose the mountains.

Can you identify the fraction of the pie chart for mountains?

Can you identify the percentages of the pie chart for mountains?

171

15 Data

Continued 5 people chose mountains as their favourite place.

There are 20 people represented in the pie chart. 25% of the people chose mountains. Work out 25% of 20.

Exercise 15.1 Focus 1 This is a table of how many umbrellas a shop sold in one week. Day

1

2

3

4

5

6

7

Number of Umbrellas

0

2

6

4

0

1

0

a Draw a dot plot of the data.

Remember to label the horizontal axis and the vertical axis.

b On which day were the most umbrellas sold? c How many umbrellas were sold in total during the week? 

172

15.1 Bar charts, dot plots, waffle diagrams and pie charts



This is a table of how many umbrellas the shop sold later in the year in one week. Day

1

2

3

4

5

6

7

Number of Umbrellas

3

4

6

6

7

5

6

d Draw a dot plot of the data.

e Write two sentences to describe how the umbrella sales were different in the two weeks represented by your dot plots. 1. In the first week



but in the second week

 .

2. In the first week



f

Why do you think there might be differences in the umbrella sales in the two weeks?

but in the second week

 .



173

15 Data

2 Four groups of children were asked to say their favourite flavour of ice-cream. The frequency table and the pie charts show the data. Draw lines to match the frequency tables to the correct pie charts. Favourite flavour Vanilla

1

Strawberry

1

Chocolate

1

Lemon

1

Favourite flavour

Frequency

Vanilla

2

Strawberry

0

Chocolate

0

Lemon

2

Favourite flavour

Frequency

Vanilla

0

Strawberry

3

Chocolate

1

Lemon

0

Favourite flavour

174

Frequency

Frequency

Vanilla

4

Strawberry

0

Chocolate

0

Lemon

0

15.1 Bar charts, dot plots, waffle diagrams and pie charts

Practice 3 This table shows the number of children in each household along a street. a Complete the table with the percentages of households in each category. Number of children

0

1

2

3

4

Frequency

3

2

3

1

1

Percentage b Complete the pie chart and waffle diagrams to represent the data.

Key



   

c What percentage of the households had children? d Is it easier to see the percentage of households that have children from the table, the pie chart or the waffle diagram? Explain your answer.

175

15 Data

4 20 people were asked to respond to this survey question – Are the school holidays too short? This is the pie chart made from the data collected. Mostly disagree 20% Mostly agree 55%

Strongly disagree 10% Don’t know 5% Strongly agree 10%



Convert the data from the pie chart into a bar graph. Use this space for your working.



Use the data from the pie chart or your bar graph to write three true statements. 1. 2. 3.

176

15.1 Bar charts, dot plots, waffle diagrams and pie charts

Challenge 5 Juma surveyed people outside a swimming pool and outside a cinema about their favourite activities. These waffle diagrams represent her data.

Waffle Diagram A Key reading watching TV playing computer games swimming running



Waffle Diagram B Key reading watching TV playing computer games swimming running



177

15 Data

a Which waffle diagram do you think represents the survey taken outside the swimming pool? Why? b What percentage of people in waffle diagram B prefer playing computer games? c What percentage of people in waffle diagram A do not prefer watching TV? d If 50 people are represented in waffle diagram A. How many of those people prefer watching TV? e Describe one thing that is similar between the two sets of data represented in the waffle diagrams. f

If you carried out the same survey with people in your class what do you predict the data would look like in a waffle diagram? Why? Describe how you predict it might be similar or different to another class in your school.

g Describe how you would conduct an investigation to find out the favourite activities of two classes in your school to check your predictions.

178

15.1 Bar charts, dot plots, waffle diagrams and pie charts

6 The Tornadoes and the Hurricanes football teams played in a tournament.

The mode of the Tornadoes goals scored was 1.



The range of the goals scored by the Hurricanes was 4.



Sort these 6 different graphs and charts. There are 3 representations of the games won and goals scored by each of the two different teams. A Bar chart of goals scored

Number of games

6 5 4 3 2 1 0 0



1

2

3

4

5

Number of goals

B Waffle diagram of goals scored Key 0 1 2 3 4 5



179

15 Data

C

Dot plot of goals scored 6

Number of games

5 4 3 2 1 0 0

1

2

3

4

5

Number of goals





D Pie chart of the outcomes of the games

Won Lost Drew

E Tally chart of the outcomes of the games Game outcome

180

Tally

Total

Won

IIII IIII I

11

Lost

IIII II

7

Drew

II

2

15.1 Bar charts, dot plots, waffle diagrams and pie charts

F Venn Diagram

Games where the team scored more than 1 goal

Games where the team won

4

7

4

5



Write the 3 letters of the representations that match each team.



The Tornadoes

 

 



The Hurricanes

 

 

181

15 Data

15.2 Frequency diagrams, line graphs and scatter graphs Worked example 2

frequency diagram

Hours playing sport

This scatter graph shows how many hours 10 people play sport in a week and how quickly they run 10 km. 20 18 16 14 12 10 8 6 4 2 30

35 40 45 50 55 Time to run 10 km (min)

line graph scatter graph

60

Hours playing sport

Use the line of best fit to estimate how many hours of sport someone does if they run 10 km in 55 minutes. 20 18 16 14 12 10 8 6 4 2 30

182

Time to run 10 km is on the horizontal axis. Find 55 minutes on the horizontal axis and follow the find up.

35 40 45 50 55 Time to run 10 km (min)

60

15.2 Frequency diagrams, line graphs and scatter graphs

Hours playing sport

Continued 20 18 16 14 12 10 8 6 4 2 30

Where the line from 55 minutes meets the line of best fit, look along to find how high it is on the vertical axis. It is at 5 hours on the vertical axis.

35 40 45 50 55 Time to run 10 km (min)

60

We can estimate that someone who runs 10 km in 55 minutes plays sport for 5 hours a week.

Exercise 15.2 Focus 1 These are the heights of a group of children in centimetres.

129 145

135 146 128 151 140 136 141 142



125 134

147 150 144 131 152 133 136 146



141 147

137 148 153 145 128 149 150 147



Complete the table below.

183

15 Data



Tally the heights of the children and write the totals. Height (cm)

Tally

Total

125 – less than 130 130 – less than 135 135 – less than 140 140 – less than 145 145 – less than 150 150 – less than 155



184

Draw a frequency diagram on the squared paper to show the children’s heights.

15.2 Frequency diagrams, line graphs and scatter graphs

2 Sofia’s height was measured every year from when she was born until she was 12 years old. This line graph represents the data collected. 160 140

Height (cm)

120 100 80 60 40 20 0

0

1

2



3

4

7 5 6 Age (years)

8

9

10

11

12

a How tall was Sofia when she was 4 years old? b What age was Sofia when she was 130 cm tall? c Use the line graph to estimate Sofia’s height when she was 5

1 2

years old.

d What happened to Sofia’s height between the ages of 10 and 11 years? How do you know?

185

15 Data

3 Look at the lines of best fit on these scatter graphs. Are they correct? Write wrong direction, too high, too low, too steep, not steep enough, or just right to describe the line. a



b







c

d











e

f



186







15.2 Frequency diagrams, line graphs and scatter graphs

Practice 4 These two frequency diagrams show the mass of a group of boys and a group of girls. Boys

Frequency

6 5 4 3 2 1 0 25

30

35

40

45

50

55

Mass (kg)

Girls

Frequency

6 5 4 3 2 1 0 25



30

35

40

45

50

55

Mass (kg)

a How many boys are between 45 kg and 50 kg? b How many girls have a mass 40 kg or greater?

187

15 Data

c Tom says: ‘The child with the greatest mass is a boy.’

Circle your reply.



• Tom is correct.



• Tom is not correct.



• Tom could be correct, but we cannot tell from the graphs.



Explain your reply.

5 Imagine you investigate the mass of fruit and vegetables and put the data into two frequency diagrams.

a What equipment would you need to conduct the investigation?

188

15.2 Frequency diagrams, line graphs and scatter graphs

b Draw a table that you could use to collect the data.

c What do you predict that the data would show? Why? 6 The ages and heights of 24 palm trees are recorded in this table. a Complete the scatter graph to represent the data. Remember to label the axes. Age of tree (years)

Age of tree (years)

Height of tree (m)

3.5

17

6

9

2

21

9

1

0.5

21

8.5

10

Height of tree (m)

15

5

2

0.5

3

1

8

4

6

3

17

7

5

2

14

6

5

1.5

12

3

5

24

9.5

3.5

13

5

22

7

15

4.5

25

11

15

7

17 9

189

15 Data

12 10 8 6 4 2



0

0

5

10

15

20

25

30

b Draw a line of best fit on the scatter graph. c Use your line of best fit to estimate the height of a tree that is 11 years old. d Use your line of best fit to estimate the age of a tree that is 8 m tall. Challenge 7 This line graph shows temperatures recorded every two hours over a 24 hour period.

Temperature (°C)

6 4 2 0 –2 –4 12

2

midnight

4

6

8

10 12

2

4

6

8

noon Time



a What happened to the temperature between 6pm and 8pm?

190

10 12 midnight

15.2 Frequency diagrams, line graphs and scatter graphs

b Use the line graph to estimate the temperatures at these times. i 3pm ii 9pm iii 9am c Why can we only use the graph to estimate the temperature at 3pm, and not know the temperature precisely?



14 14 15 16 18 20 21 22 20 18 16 15 14

6

8

10

2

4

6

8

10

Represent the data in a line graph.

12

2

midnight



4

12 midnight

Temperature (oC)

2

12 noon

Time

12 midnight

d This table also shows temperatures recorded every two hours over a 24 hour period.

4

6

8

10 12 noon

2

4

6

8

10 12 midnight

Time

191

15 Data

e Describe what is similar and what is different about the two line graphs. f

Give two possible explanations for the difference between the graphs 1. 2.

g If you recorded the temperatures outside where you live every two hours for 24 hours what do you predict would be similar and different about your line graph to the line graphs above. Why? 8 Some people attended a course. At the end of the course there was an exam.

192

This table shows how many days of the course a person did not attend and the percentage of correct marks they scored on the exam. Days absent

0

1

0

2

5

3

1

4

7

2

Percentage scored

85

88

92

74

59

42

67

50

48

57

Days absent

5

3

7

1

4

3

0

2

3

5

Percentage scored

36

61

23

85

64

65

70

70

48

42

15.2 Frequency diagrams, line graphs and scatter graphs

a Draw scatter graph of the data in the table, including a line of best fit.

b Use your line of best fit to estimate the percentage that would be score by someone who was absent for 6 days of the course. c Describe the link between the number of days absent and the percentage scored on the test.

193

16 The laws of ­arithmetic 16.1 The laws of arithmetic Worked example 1

associative rule  brackets

Put brackets in the calculation to make it correct.

commutative rule   distributive rule order of operations

4 + 5 × 3 × 2 = 54 4 + 5 × 3 × 2 = 4 + 30

= 34

The order of operations is multiplication before addition.

Which is not the required answer. (4 + 5) × 3 × 2 = 9 × 3 × 2

= 54

Answer: (4 + 5) × 3 × 2 = 54

194

Brackets are worked out before multiplication.

16.1 The laws of arithmetic

Exercise 16.1 Focus 1 Work out the answer to each calculation.

Join each calculation to the correct answer. The first one has been done for you.



2 Which expression has the same value as 9 × (6 – 1)?

Calculation

Answer

(12 – 3) × 8

18

10 × 8 + 1

27

6 × (5 – 2)

36

7 × (4 + 5)

45

6 × (3 + 6)

54

(9 – 4) × 9

63

(8 + 4) × 3

72

(12 – 3) × 3

81

54 – 5   54 – 1   9 × 5   9 × 7

3 Work out the answer to each question to help you find your way through the maze. start

3 × (14 – 6)

24

36

10 ÷ (8 – 3)

13

4

15

2

9 × (18 – 9)

20 ÷ (2 + 3)

5 × (3 + 4)

25

35

19

81

20 ÷ (7 – 3)

6 × (13 + 2)

24 ÷ (3 + 5)

3

3

end



195

16 The laws of ­arithmetic

4 Safiya is learning to use brackets.

She writes down four calculations but one of her calculations is wrong.



A: 10 + (2 + 8) × 3 = 40

B: (10 + 2 + 8) × 3 = 60



C: (10 + 2) + 8 × 3 = 58

D: 10 + 2 + 8 × 3 = 36



Which calculation is wrong? What is the correct answer?

Practice 5 Complete the calculation.

36 × 97 = 36 × (100 –







=



=

 )

= (36 × 100) – (36 ×

 )

–

6 Mandy writes 4 + 9 × 5 × 2 = 130

Is she correct?



Explain your answer.

7 Write the missing number to make the calculation correct.

196

(10 –

 ) × 10 = 10

16.1 The laws of arithmetic

8 Which number sentence is equivalent to 8 × 12?



A: (8 × 10) + (8 × 2)

B: (8 × 1) + (8 × 2)

C: (8 × 10) + 2

D: 8 + (10 × 2)

Explain why the other sentences are wrong.

9 Write the sign or = to make each expression correct. a 2 × (3 + 4)

2×3+4

b (10 + 6) ÷ 2

5+6÷2

c (13 + 5) ÷ 9

(13 + 8) ÷ 7

Challenge 10 Use the numbers 2, 3, 4 and 5 to complete the calculations. a 20 – ( 

+ 9) = 7

b 9 +

c 11 – ( 

– 3) = 9

d 11 –

–3=8 + 2 = 10

11 Draw brackets to make each expression equal to 10. a 14 – 12 × 5

b 11 – 6 – 5

c 20 – 15 – 5

d 20 ÷ 4 – 2

197

16 The laws of ­arithmetic

12 Write the missing numbers. a 3 × ( 

+ 4) = 27

c (6 + 7) × e 3 × ( 

= 39

b 4 × (8 –

 ) = 20

d 16 ÷ ( 

– 3) = 16

f

– 6) = 6

(21 – 12) ÷

=3

13 Here are six cards.

7

8

25

(

)

=

Use all these cards together with any of the operation signs +, –, × and ÷ to make a number sentence with an answer of 10.

14 Use these numbers together with brackets and operation signs to make the target number.

198

Example: 3, 4, 6

Target 42

a 2, 5, 6

Target 40

b 2, 3, 5

Target 4

c 3, 4, 6

Target 12

Answer (3 + 4) × 6

17

Transformations

17.1 Coordinates and translations Worked example 1

axes   axis

Translate rectangle ABCD 2 squares right and 4 squares up. Label it A’B’C’D’. Write down the coordinates of the vertices A’, B’, C’ and D’.

corresponds  translate

y 5 4 3 2 1 0 –5 –4 –3 –2 –1 –1 A B –2 D

1

2

3

4

5

x

–3 C –4 –5

199

17 Transformations

Continued Move each vertex of the rectangle 2 squares right and 4 squares up.

y 5 4 A'

3

B'

2 D'

1

0 –5 –4 –3 –2 –1 –1 A B –2 D

–3 C –4 –5

Aˊ (-2, 3) Bˊ (1, 3) Cˊ (1, 1) Dˊ (–2, 1)

200

C' 1

2

3

4

5

x

Vertex A on the original corresponds with vertex Aˊ (you say A dash) on the translated rectangle. B corresponds with Bˊ, C corresponds with Cˊ and D corresponds with Dˊ. Remember that when you write coordinates the x-axis number is first and the y-axis number is second.

17.1 Coordinates and translations

Exercise 17.1 Focus 1 Match each point on the grid to its correct coordinates. y 5 4

A

3 B

2 1

0 –5 –4 –3 –2 –1 –1

1

2

3

4

5

x

–2 –3 D

C

–4

Tip

–5

(+, +) (right, up)

i

(2, –3)

iii (2, 3)

ii (–3, 1) iv (–3, –4)

(+, -) (right, down) (-, +) (left, up) (-, -) (left, down)

201

17 Transformations

2 Here is an orienteering map. It shows the start and finish, and checkpoints A, B, C and D on the route. y 4 START/ FINISH

3 D 2 1

–5

–6

–4

–3

–2

–1

A

0

1

–1

2

3

4

5

–2 B –3 –4

C



Write down the coordinates of checkpoints A, B, C and D.



A





B



C

3 The diagram shows four triangles. Tip

y 6 5 B

4 3

T

2 1



202

0

Start with triangle T and work out how you can move T to A, T to B, and T to C.

A

C 1

2

3

4

5

6

x

D

6

x

17.1 Coordinates and translations



Triangle T has been translated to triangles A, B and C.



Match each translation i, ii and iii to the correct triangle A, B and C. i

1 square right and 2 squares down

ii 2 squares right and 3 squares up iii 1 square left and 2 squares up. Practice 4 The diagram shows line segments PQ and RS.

Write down the coordinates of the points P, Q, R and S. y 5 4 3 2 P

Q

1

0 –5 –4 –3 –2 –1 –1

1

2

3

4

x

5 R

–2 S

–3 –4 –5



P



Q



R



S

203

17 Transformations

5 Draw axes from –5 to +5 on squared paper. Draw a parallelogram with vertices at E (–2, –2), F (2, –2), G (4, –4) and H (0, –4).

a Translate parallelogram EFGH 1 square right and 6 squares up. Label the parallelogram EˊFˊGˊHˊ and write down the coordinates of its vertices.

Eˊ



Fˊ



Gˊ



Hˊ

b Translate parallelogram EFGH 3 squares left and 1 square down. Label the parallelogram EˊˊFˊˊGˊˊHˊˊ and write down the coordinates of its vertices.

204

Eˊˊ



Fˊˊ



Gˊˊ



Hˊˊ

17.1 Coordinates and translations

6 Draw axes from –6 to +6 on squared paper. Plot the points A (0, 1), B (2, 1) and C (4, –2).

a W  rite down the coordinates of D so that A, B, C and D are the vertices of an isosceles trapezium. b W  rite down two possible coordinates of D so that D is a point on the line segment AB.

Tip You can use fractions or decimals in coordinates.

205

17 Transformations

c Write down two possible coordinates of D so that A, B, C and D are the vertices of a parallelogram.





d Is it possible to find coordinates for D so that A, B, C and D are the vertices of a rectangle? Explain your answer. Challenge 7 (-1, 3) and (3, 1) are the coordinates of two vertices of a square. What could the other vertices of the square be? Find all the possible solutions. 8 The diagram shows shape P on a coordinate grid. y 6 5 4 3 P



206

–5 –4 –3 –2 –1

2 1 0

1

2

3

Erin translates shape P 2 squares right and 3 squares up. She labels the shape Q.

4

5

x

17.1 Coordinates and translations

a What translation should Erin do to take shape Q back to shape P? Explain how you worked out your answer. b Erin translates shape Q 3 squares right and 1 square up. She labels the shape R. i

Erin thinks that she could use the single translation 6 squares right and 4 squares up to take shape P to shape R. Is Erin correct? Explain your answer.

ii What do you notice about the single translation P to R, and the two translations P to Q and Q to R? 9 Rectangle K has vertices as the points (-2, -1), (-5, -1), (-5, -3) and (-2, -3).

Shen translates K four times, using these four different translations A, B, C and D. A 3 squares right and 2 squares up

B 5 squares right and 6 squares up

C 3 squares right

D 1 square left and 1 square down

207

17 Transformations



After which translation will K and the new rectangle be: a touching end to end b touching corner to corner c overlapping d not touching or overlapping?

17.2 Reflections Worked example 2

diagonal mirror line

Reflect this triangle in the diagonal mirror line.

Take one vertex of the triangle at a time. Draw arrows (black) to the mirror line, then draw the same length arrows (grey) the other side of the mirror line. Join the vertices with straight lines to complete the reflected triangle.

208

17.2 Reflections

Exercise 17.2 Focus 1 Which drawings show correct reflections of triangle A? a

b A A

c

d A A



209

17 Transformations

2 Reflect each shape in the mirror lines. They have all been started for you. a

b

c

d

3 Is A, B or C the correct reflection for each of these? i



A

210

B

C

17.2 Reflections

ii



A

iii

B

C

B

C



A

211

17 Transformations

Practice 4 Reflect the shape in the horizontal and vertical mirror lines. a

b

c

d

5 This is part of Jose’s homework.



Question: Reflect shape A in the diagonal line of symmetry.



Label your answer shape B.

B A



212

Has Jose drawn shape B correctly? Explain your answer.

17.2 Reflections

6 Reflect the shape in the diagonal mirror lines. a

b

c

d

7 a Describe the mirror line for each of these reflections. i

ii





iii Tip Is the mirror line horizontal, vertical or diagonal? b Draw in the correct mirror line for each reflection.

213

17 Transformations

Challenge 8 Draw in the correct mirror line for each reflection. a

b

c

9 a Reflect the shapes in the mirror lines to complete the pattern. i

ii

b What is the order of rotational symmetry of the completed pattern? i

ii

10 The diagram shows shape A on a coordinate grid.

A

a Reflect shape A in the mirror line. Label the new shape B. b Translate shape B 2 squares left and 1 square down. Label the new shape C. c Reflect shape C in the mirror line. Label the new shape D. d Describe the translation that takes shape D back to shape A. e What do you notice about your answer to part d and the translation you carried out in part b?

214

17.3 Rotations

17.3 Rotations Worked example 3

anticlockwise   centre of rotation

Rotate triangle A 90° anticlockwise about the centre of rotation marked C. Label your answer triangle B.

clockwise  corresponding  rotate

A C

Step 2

Step 1 A

A

C

C

Trace the shape, then put your point of your pencil on the centre of rotation.

Start turning the tracing paper 90° (a quarter turn) anticlockwise.

Step 3

Step 4 A A

B

C

C

Once the turn is completed make a note of where the new triangle is.

Draw the new triangle onto the grid and label it B.

215

17 Transformations

Exercise 17.3 Focus 1 In each of these diagrams, shape A has been rotated to shape B around centre C. Write down if the rotation is clockwise or anticlockwise. a

b

A B

C

Tip

A C

B



Clockwise: 



c

d

B C

A

B A



C

Anticlockwise:



2 Complete these rotations of 90° clockwise about the centre C. a

b

C

C

3 Complete these rotations of 90° anticlockwise about the centre C. a

C

b

C

Practice 4 Rotate the shapes 90° clockwise about the centre C. a

b C

216

c C

C

17.3 Rotations

5 Rotate the shapes 90° anticlockwise about the centre C. a

b

c

C

C C

6 This is part of Alysha’s homework. The centre of rotation is shown by a dot (•).



Question:



Rotate shape A 90° clockwise about the centre of rotation (•). Label the shape B.



Answer: 



Has Alysha got her homework correct? Use diagrams to help you explain your answer.

A

B

Challenge 7 Rotate the shapes 90° about the centre of rotation C, using the direction shown. a

  anticlockwise

b

C

  clockwise C

8 a Follow these instructions to make a pattern.      

1. Rotate the shape 90° clockwise about C.



2. Draw the new shape.



3. Rotate the new shape 90° clockwise about C.

C

217

17 Transformations



4. Draw the new shape.



5. Rotate the new shape 90° clockwise about C.



6. Draw the new shape.

b What is the order of rotational symmetry of your completed pattern? 9 a Rotate triangle A1B1C, using the same instructions as question 8a.      

Label the vertices of the three new triangles A2, B2, C then A3, B3, C then A4, B4, C.

A1 C

B1

b On your completed diagram, join A1 to A2 to A3 to A4 to A1 with straight lines. What shape have you just drawn? c On your completed diagram, join B1 to B2 to B3 to B4 to B1 with straight lines. What shape have you just drawn? d Do you think that whatever shape you rotate, if you rotate it 90° clockwise or anticlockwise three times, then the shape you get when you join corresponding vertices will always be the same? Explain your answer. You can use diagrams to help your explanation.

218

17.3 Rotations

Acknowledgements

219