Greg Byrd, Lynn Byrd and Chris Pearce

Cambridge Checkpoint

Mathematics

Practice Book

9

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK www.cambridge.org Information on this title: www.cambridge.org/9781107698994 © Cambridge University Press 2013 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 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library ISBN 978-1-107-69899-4 Paperback Cover image © Cosmo Condina concepts / Alamy 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.

Contents Introduction 1 Integers, powers and roots 1.1 Directed numbers 1.2 Square roots and cube roots 1.3 Indices 1.4 Working with indices

5 7 7 8 9 10

9.6 9.7

10 Processing and presenting data 10.1 Calculating statistics 10.2 Using statistics

48 48 50

2 2.1 2.2 2.3

Sequences and functions Generating sequences Finding the nth term Finding the inverse of a function

11 11 12 13

3 3.1 3.2 3.3 3.4

Place value, ordering and rounding Multiplying and dividing decimals mentally Multiplying and dividing by powers of 10 Rounding Order of operations

14 14 15 16 17

11 11.1 11.2 11.3 11.4

Percentages Using mental methods Comparing different quantities Percentage changes Practical examples

52 52 53 54 55

4 4.1 4.2 4.3

Length, mass, capacity and time Solving problems involving measurements Solving problems involving average speed Using compound measures

18 18 19 20

12 12.1 12.2 12.3 12.4 12.5

Tessellations, transformations and loci Tessellating shapes Solving transformation problems Transforming shapes Enlarging shapes Drawing a locus

56 56 57 59 60 61

5 5.1 5.2 5.3 5.4 5.5 5.6

Shapes Regular polygons More polygons Solving angle problems Isometric drawings Plans and elevations Symmetry in three-dimensional shapes

21 21 22 23 25 26 27

13 13.1 13.2 13.3 13.4 13.5 13.6

Equations and inequalities Solving linear equations Solving problems Simultaneous equations 1 Simultaneous equations 2 Trial and improvement Inequalities

62 62 63 64 65 66 67

6 6.1 6.2 6.3 6.4

Planning and collecting data Identifying data Types of data Designing data-collection sheets Collecting data

28 28 29 30 31

14 Ratio and proportion 14.1 Comparing and using ratios 14.2 Solving problems

68 68 70

7 7.1 7.2 7.3 7.4 7.5

Fractions Writing a fraction in its simplest form Adding and subtracting fractions Multiplying fractions Dividing fractions Working with fractions mentally

32 32 33 34 35 36

15 15.1 15.2 15.3 15.4

Area, perimeter and volume Converting units of area and volume Using hectares Solving circle problems Calculating with prisms and cylinders

72 72 73 74 75

16 16.1 16.2 16.3

Probability Calculating probabilities Sample space diagrams Using relative frequency

76 76 77 78

8 8.1 8.2 8.3

Constructions and Pythagorasâ&#x20AC;&#x2122; theorem Constructing perpendicular lines Inscribing shapes in circles Using Pythagoras' theorem

37 37 38 39

17 Bearings and scale drawings 17.1 Using bearings 17.2 Making scale drawings

80 80 82

9 9.1 9.2 9.3 9.4 9.5

Expressions and formulae Simplifying algebraic expressions Constructing algebraic expressions Substituting into expressions Deriving and using formulae Factorising

40 40 41 43 44 45

18 18.1 18.2 18.3 18.4 18.5 18.6

83 83 85 86 87 89 90

Adding and subtracting algebraic fractions Expanding the product of two linear expressions

Graphs Gradient of a graph The graph of y = mx + c Drawing graphs Simultaneous equations Direct proportion Practical graphs

46 47

3

19 Interpreting and discussing results 19.1 Interpreting and drawing frequency diagrams 19.2 Interpreting and drawing line graphs 19.3 Interpreting and drawing scatter graphs 19.4 Interpreting and drawing stem-andleaf diagrams 19.5 Comparing distributions and drawing conclusions

4

91 91 92 93 94 95

Introduction

Welcome to Cambridge Checkpoint Mathematics Practice Book 9

The Cambridge Checkpoint Mathematics course covers the Cambridge Secondary 1 Mathematics framework. The course is divided into three stages: 7, 8 and 9. This Practice Book can be used with Coursebook 9. ItÂ is intended to give you extra practice in all the topics covered in the Coursebook. Like the Coursebook, the Practice Book is divided into 19 units. In each unit you will find an exercise for every topic. These exercises contain similar questions to the corresponding exercises in the Coursebook. This Practice Book gives you a chance to try further questions on your own. This will improve your understanding of the subject. It will also help you to feel confident about working on your own when there is no teacher available to help you. There are no explanations or worked examples in this book. If you are not sure what to do, or need to remind yourself about something, look at the explanations and worked examples in the Coursebook.

5

1

Integers, powers and roots

Exercise 1.1

Directed numbers

1 Work these out. a −6 + 2.7 c 16 + −2.7

b −6 + −2.7 d 2.7 + −16

2 Work these out. a 7 − −5 c −7.1 − −5.2

b 7.1 − −5.2 d −5.2 − −7.1

3 Work these out. a −8.4 + 12.1 c 8.4 − −12.1

b −8.4 − 12.1 d −12.1 − −8.4

4 These are five temperatures, in degrees Celsius (°C). 1.5

−3.5

−7

−10

−3

Find the mean temperature. 5 Solve these equations. a N + 2.3 = −4.7 b 2N + 6.8 = −10.2 c N ÷ 4 = −2.7 6 Work these out. a −2 × 3.4 c −3 × 9.2

b −4.8 ÷ −4 d 14 ÷ −4

7 Copy and complete this multiplication table. × −1.1

−1.2

3 −1.5

8 Use the information in the box to work out the value of each expression. a r+s+t b (r − s) − t r = 8.4 s = 6.4 t = −7.4 c (s − r) × t d t ÷ (r − s) e (r + s) ÷ t 9 A + B = 0 and AB = −36. What is the value of A − B?

1

Integers, powers and roots

7

Exercise 1.2

Square roots and cube roots

1 Estimate each root, to the nearest whole number. b 150 a 50 c

350

d

3

350

2 Explain why: a 95 must be between 9 and 10. b 3 95 must be between 4 and 5. 3

3 < 10.5 < 4

Write a similar statement for each of these roots. b 3 500 a 385 c 4 a

69.8

d

3

55.5

144 < N < 225

What can you say about N ? b

c

100 < M < 400

What can you say about M ? 0 < R < 125

What can you say about 3 R ? 5

25.52 = 650.25

26.52 = 702.25

a Estimate 690 to the nearest whole number. b Estimate 650 to one decimal place. c Estimate 700 to one decimal place. 6 Show that 3 200 is less than half 200. 7 a Show that 7500 is more than 80. b Show that 3 7500 is less than 20. 8 Use a calculator to find the following square roots. a 30.25 b 441 c e

841 174.24

d

54.76

9 Use a calculator to find the following square roots. Round your answers to two decimal places. b 60 a 6 c e 8

600 2.43

d

42.65

1 Integers, powers and roots

Do not use a calculator in this exercise, except for questions 8 and 9.

Exercise 1.3 Indices 1 Write each number as an integer or a fraction. a 54 b 35 c 6−2 d 2−3 e 40 2 Write each number as a decimal. a 8−1 b 2−2 c 4−1 d 3−1 e 10−3 3 Write these numbers in order of size, smallest first. 112 26 34 43 62 121

4 Write these numbers in order of size, smallest first. 1−5 2−4 3−3 4−2 5−1

5 Write each number as a power of 4. a 16 b 256 c 1 d 14 e 1 64 6 Write each of the numbers in question 5 as a power of 2. 7 3N = 9−2 Work out the value of N. 8 Write each expression as a single number. a 3−1 + 6−1 b 42 + 41 + 40 + 4−1 + 4−2

1

Integers, powers and roots

9

Exercise 1.4

Working with indices

1 Simplify each expression. Write the answers in index form. a 83 × 82 b 7 × 73 c 22 × 22 × 22 d r2 × r4 e s3 × s2 × s 2 Simplify each expression. a 42 × 41 b 6 × 60 c c2 × c 2 d 12 × 25 e e × e0

3 Simplify each expression in the box. One is different from the other four. Which one? a4 × a2 a 5 × a a6 × a0 a0 × a5 a3 × a3

Give a reason for your answer. 4 This table shows powers of 9. 91 9

92 81

93 729

94 6561

95 59 049

Use the table to find the value of each expression. a

531 441

b

3

531 441

5 Simplify each expression, writing it as a single power. a a5 ÷ a3 b 6 ÷ 63 c 82 ÷ 8 d d 2 ÷ d2 e e ÷ e2 6 Write each of these as a fraction. b k 2 ÷ k3 a 32 ÷ 34 −4 c 10 × 10 × 10 d 42 ÷ 25 7 Simplify each expression. 3 2 3 3 a a ×a b 5 ×5 a 5 3 4 f ×f 2 c d 10 2× 10 f 10 × 10 8 Find the value of n in each equation. b 10n ÷ 10 = 0.1 a 5n × 53 = 625 0 2 c n × n × n = 64

10

1 Integers, powers and roots

96 531 441