Cambridge Checkpoint Mathematics Practice Book 8 matches the requirements of stage 8 of the revised Cambridge Secondary 1 curriculum framework. It is endorsed by Cambridge International Examinations for use with their programme. The series is written by an author team with extensive experience of both teaching and writing for secondary mathematics. This supportive Practice Book is intended to be used alongside the Cambridge Checkpoint Mathematics Coursebook 8. It contains exercises that will help students develop the skills they need to succeed with the Secondary 1 programme. The Practice Book: r contains exercises for each Coursebook topic, arranged in the same order as the topics in the Coursebook r allows students to practise using the concepts they have learned and develop their problem-solving skills. Answers to the exercises are included on the Teacher’s Resource 8 CD-ROM. Other components of Cambridge Checkpoint Mathematics 8: Coursebook 8 Teacher’s Resource 8
Greg Byrd, Lynn Byrd and Chris Pearce
Cambridge Checkpoint
Mathematics
Practice book
Practice Book 8
ISBN 978-1-107-69787-4 ISBN 978-1-107-62245-6
Completely Cambridge – Cambridge resources for Cambridge qualifications Cambridge University Press works closely with Cambridge International Examinations as parts of the University of Cambridge. We enable thousands of students to pass their Cambridge exams by providing comprehensive, high-quality, endorsed resources. To find out more about Cambridge International Examinations visit www.cie.org.uk Visit education.cambridge.org/cie for information on our full range of Cambridge Checkpoint titles including e-book versions and mobile apps.
Byrd, Byrd and Pearce
9781107665996 Byrd, Byrd & Pearce: Cambridge Checkpoint Mathematics Practice Book 8 Cover. C M Y K
Greg Byrd, Lynn Byrd and Chris Pearce
Cambridge Checkpoint Mathematics
Cambridge Checkpoint Mathematics Practice Book 8
8
Greg Byrd, Lynn Byrd and Chris Pearce
Cambridge Checkpoint
Mathematics
Practice Book
8
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/9781107665996 © 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 in the United Kingdom by Latimer Trend A catalogue record for this publication is available from the British Library ISBN 978-1-107-66599-6 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 Arithmetic with integers 1.2 Multiples, factors and primes 1.3 More about prime numbers 1.4 Powers and roots
5 7 7 8 9 10
2 2.1 2.2 2.3 2.4 2.5 2.6
Sequences, expressions and formulae Generating sequences Finding rules for sequences Using the nth term Using functions and mappings Constructing linear expressions Deriving and using formulae
11 11 12 13 14 15 16
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
Place value, ordering and rounding Multiplying and dividing by 0.1 and 0.01 Ordering decimals Rounding Adding and subtracting decimals Dividing decimals Multiplying by decimals Dividing by decimals Estimating and approximating
17 17 18 19 20 21 22 23 24
4 4.1 4.2
Length, mass and capacity Choosing suitable units Kilometres and miles
25 25 26
5 5.1 5.2 5.3
Angles Parallel lines Explaining angle properties Solving angle problems
27 27 29 30
6 6.1 6.2 6.3
Planning and collecting data Collecting data Types of data Using frequency tables
31 31 33 34
7 7.1
36
7.2 7.3 7.4 7.5 7.6 7.7 7.8
Fractions Finding equivalent fractions, decimals and percentages Converting fractions to decimals Ordering fractions Adding and subtracting fractions Finding fractions of a quantity Multiplying an integer by a fraction Dividing an integer by a fraction Multiplying and dividing fractions
8 8.1 8.2
Shapes and geometric reasoning Recognising congruent shapes Identifying symmetry of 2D shapes
43 43 44
36 37 38 39 40 41 41 42
8.3 8.4 8.5
Classifying quadrilaterals Drawing nets of solids Making scale drawings
45 46 47
9
Simplifying expressions and solving equations Collecting like terms Expanding brackets Constructing and solving equations
49 49 50 52
9.1 9.2 9.3
10 Processing and presenting data 10.1 Calculating statistics from discrete data 10.2 Calculating statistics from grouped or continuous data 10.3 Using statistics to compare two distributions
53 53
11 11.1 11.2 11.3 11.4
Percentages Calculating percentages Percentage increases and decreases Finding percentages Using percentages
56 56 57 58 59
12 12.1 12.2 12.3 12.4
Constructions Drawing circles and arcs Drawing a perpendicular bisector Drawing an angle bisector Constructing triangles
60 60 61 62 64
13 13.1 13.2 13.3 13.4
Graphs Drawing graphs of equations Equations of the form y = mx + c The midpoint of a line segment Graphs in real-life contexts
65 65 66 67 68
14 14.1 14.2 14.3
Ratio and proportion Simplifying ratios Sharing in a ratio Solving problems
70 70 71 73
54 55
15 Probability 15.1 The probability that an outcome does not happen 15.2 Equally likely outcomes 15.3 Listing all possible outcomes 15.4 Experimental and theoretical probabilities
74
16 Position and movement 16.1 Transforming shapes 16.2 Enlarging shapes
79 79 81
17 Area, perimeter and volume 17.1 The area of a triangle 17.2 The areas of a parallelogram and trapezium
84 84 84
74 75 76 77
3
17.3 17.4 17.5 17.6
The area and circumference of a circle The areas of compound shapes The volumes and surface areas of cuboids Using nets of solids to work out surface areas
18 Interpreting and discussing results 18.1 Interpreting and drawing frequency diagrams 18.2 Interpreting and drawing pie charts 18.3 Interpreting and drawing line graphs 18.4 Interpreting and drawing stem-and-leaf diagrams 18.5 Drawing conclusions
4
86 87 88 89 90 90 91 93 94 95
Introduction
Welcome to Cambridge Checkpoint Mathematics Practice Book 8
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 8. It is intended to give you extra practice in all the topics covered in the Coursebook. Like the Coursebook, the Practice Book is divided into 18 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
Arithmetic with integers
1 Add these numbers. a 6 + −3 b −6 + −4
c −2 + −8
2 Find the missing integer in each case. b 4 + = −6 a 5+ =2 e 7 + = −6 d −12 + = −8
d −1 + 6 c −3 +
3 Subtract. a 3−7
b −3 − 7
c −20 − 30
4 Subtract. a 4 − −6 d −6 − −12
b 10 − −3 e 15 − −10
c −10 − −5
e −10 + 4
=3
d 5 − 15
e −9 − 4
Add the inverse.
5 In each wall diagram, add the two numbers above to get the number below. For example, 3 + −5 = −2. Find the bottom number in each diagram. b c a –5
3
4
–4
2
–1
–1
4
–6
–2
6 Copy this multiplication table. Fill in the missing numbers.
×
−3
−1
2
5
−3 −1 2 5
7 Complete these divisions. a 20 ÷ −2 b −24 ÷ 3
25
c −44 ÷ −4
d 28 ÷ −4
8 Look at the multiplication in the box. Use the same integers to write down two divisions. 9 Xavier has made a mistake. Correct it.
e −12 ÷ −6
−5 × 6 = −30
5 times 5 is 25. So −5 times −5 is −25.
10 The product of two different integers is −16. What could they be? 11 Find the missing numbers. b 4× a −2 × = 20
= −12
c
× 9 = −45
d
× −5 = −35
1
Integers, powers and roots
7
Exercise 1.2
Multiples, factors and primes
1 Find the first three multiples of each number. a 12 b 15 c 32 2 From the numbers in the box, find: a a multiple of 10 c a common factor of 27 and 36
d 50
b two factors of 24 d a prime number.
8
9
12
23
32
40
98
99
100
3 List all the prime numbers between 40 and 50. 4 Which of the numbers in the box is: a a multiple of 2 b a multiple of 5 c a common multiple of 2 and 5 d a factor of 500 e a prime number f a common multiple of 2 and 3?
95
17, 37 and 47 end in 7 and are prime numbers, so 57 and 67 must also be prime numbers.
6 Write true (T) or false (F) for each statement. a 7 is a factor of 84. b 80 is a multiple of 15. c There is only one prime number between 90 and 100. d 36 is the lowest common multiple (LCM) of 6 and 9. e 5 is the highest common factor (HCF) of 25 and 50. 7 Find the LCM of each pair of numbers. a 4 and 6 b 15 and 20 c 20 and 50
d 6 and 7
8 Find the factors of each number. a 27 b 28 c 72
d 82
e 31
9 Find the prime factors of each number. a 32 b 18 c 70
d 99
e 19
10 Find the HCF of each pair. a 12 and 15 b 12 and 18
d 12 and 25
11 The HCF of 221 and 391 is 17. Explain why 221 and 391 cannot be prime numbers. 12 Find two numbers that are not prime and have a HCF of 1.
8
1 Integers, powers and roots
97
You may use some numbers more than once.
5 Is Mia correct?
c 12 and 24
96
Exercise 1.3
More about prime numbers
1 Copy and complete these factor trees. b c a 88 135 8
11
9
15
260
26
10
2 a Draw two different factor trees for 80. b Write 80 as a product of primes. 3 Write down each number. b 24 × 33 a 2 × 32 × 52
c 22 × 112
4 84 = 22 × 3 × 7 and 90 = 2 × 32 × 5 a Write the HCF of 84 and 90 as a product of primes. b Write the LCM of 84 and 90 as a product of primes. 5 a Write each number as a product of primes. i 120 ii 160 b Find the LCM of 120 and 160. c Find the HCF of 120 and 160. 6 a Find the HCF of 84 and 96. b Find the LCM of 84 and 96. 7 a Find the HCF of 104 and 156. b Find the LCM of 104 and 156. 1000 = 23 × 53 8 10 = 2 × 5 100 = 22 × 52 Write 10 000 as a product of primes. 9
I am thinking of two prime numbers.
I can tell you their HCF.
I can tell you how to find their LCM.
a How can Sasha do that? b What will Razi tell Jake? 10 a Write 81 as a product of primes. b Write 154 as a product of primes. c Explain why the HCF of 81 and 154 must be 1.
1
Integers, powers and roots
9
Exercise 1.4
Powers and roots
1 Find the value of each of these. b 33 a 23
c 43
d 53
2 Find the value of each of these. b 34 a 24
c 44
d 104
e 10³
3 a 44 is equal to 2N. What number is N ? b 93 is equal to 3M. What number is M ? 4 The number 100 has two square roots. a What is their sum? b What is their product? 5 Find the square roots of each number. a 1 b 36 c 169
d 256
e 361
3 6 a Show that 3 −1 = 32 + 3 + 1. 2 3 4 b Show that −1 = 42 + 4 + 1. 3 c Write a similar expression involving 53.
7 The numbers in the box are all identical in value. Use this fact to write down: b 3 4096 . a 4096 8 Find the value of: b a 121 9 Find the value of: b a 38
3
289
c
125
c
3
212
400
d
27
d
10 113 = 1331. Use this fact to work out: b 3 1331 . a 114 11 Explain why Alicia is correct. A square root of 25 could be less than a square root of 16.
10
1 Integers, powers and roots
1. 3
1000
46
163
642
4096