Cambridge Lower Secondary Mathematics Learner's Book 8

Cambridge Lower Secondary Mathematics Lynn Byrd, Greg Byrd & Chris Pearce LEARNER’S BOOK 8 Second edition Digital access SAMPLE We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

Cambridge Lower Secondary Mathematics Greg Byrd, Lynn Byrd & Chris Pearce LEARNER’S BOOK 8 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

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/9781108771528 © 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 978-1-108-77152-8 Paperback with Digital Access (1 Year) ISBN 978-1-108-74642-7 Digital Edition (1 Year) ISBN 978-1-108-74639-7 eBook 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) where you are abiding by a licence granted to your school or institution by the Copyright Licensing Agency; (ii) where no such licence exists, or where you wish to exceed the terms of a licence, and you have gained the written permission of Cambridge University Press; (iii) where you are allowed to reproduce without permission under the provisions of Chapter 3 of the Copyright, Designs and Patents Act 1988, which covers, for example, the reproduction of short passages within certain types of educational anthology and reproduction for the purposes of setting examination questions. DISCLAIMER Cambridge International copyright material in this publication is reproduced under licence and remains the intellectual property of Cambridge Assessment International Education. Projects and their accompanying teacher guidance have been written by the NRICH team. NRICH is an innovative collaboration between the Faculties of Mathematics and Education at the University of Cambridge, which focuses on problem solving and on creating opportunities for students to learn mathematics through exploration and discussion https://nrich.maths.org We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

3 Introduction Welcome to Cambridge Lower Secondary Mathematics Stage 8 The Cambridge Lower Secondary Mathematics course covers the Cambridge Lower Secondary Mathematics curriculum framework and is divided into three stages: 7, 8 and 9. During your course, you will learn a lot of facts, information and techniques. You will start to think like a mathematician. This book covers all you need to know for Stage 8. The curriculum is presented in four content areas: • Number • Algebra • Geometry and measures • Statistics and probability. This book has 16 units, each related to one of the four content areas. However, there are no clear dividing lines between these areas of mathematics; skills learned in one unit are often used in other units. The book encourages you to understand the concepts that you need to learn, and gives opportunity for you to practise the necessary skills. Many of the questions and activities are marked with an icon that indicates that they are designed to develop certain thinking and working mathematically skills. There are eight characteristics that you will develop and apply throughout the course: • Specialising – testing ideas against specific criteria; • Generalising – recognising wider patterns; • Conjecturing – forming questions or ideas about mathematics; • Convincing – presenting evidence to justify or challenge a mathematical idea; • Characterising – identifying and describing properties of mathematical objects; • Classifying – organising mathematical objects into groups; • Critiquing – comparing and evaluating ideas for solutions; • Improving – Refining your mathematical ideas to reach more effective approaches or solutions. Your teacher can help you develop these skills, and you will also develop your ability to apply these different strategies. We hope you will find your learning interesting and enjoyable. Greg Byrd, Lynn Byrd and Chris Pearce Introduction We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

4 Contents Page Unit Strand of mathematics 6 How to use this book 9–28 1 Integers 1.1 Factors, multiples and primes 1.2 Multiplying and dividing integers 1.3 Square roots and cube roots 1.4 Indices Number 29–64 2 Expressions, formulae and equations 2.1 Constructing expressions 2.2 Using expressions and formulae 2.3 Expanding brackets 2.4 Factorising 2.5 Constructing and solving equations 2.6 Inequalities Algebra 65 Project 1 Algebra chains 66–79 3 Place value and rounding 3.1 Multiplying and dividing by 0.1 and 0.01 3.2 Rounding Number 80–103 4 Decimals 4.1 Ordering decimals 4.2 Multiplying decimals 4.3 Dividing by decimals 4.4 Making decimal calculations easier Number 104 Project 2 Diamond decimals 105–125 5 Angles and constructions 5.1 Parallel lines 5.2 The exterior angle of a triangle 5.3 Constructions Geometry and measure 126–136 6 Collecting data 6.1 Data collection 6.2 Sampling Statistics 137–170 7 Fractions 7.1 Fractions and recurring decimals 7.2 Ordering fractions 7.3 Subtracting mixed numbers 7.4 Multiplying an integer by a mixed number 7.5 Dividing an integer by a fraction 7.6 Making fraction calculations easier Number 171–196 8 Shapes and symmetry 8.1 Quadrilaterals and polygons 8.2 The circumference of a circle 8.3 3D shapes Geometry and measure 197 Project 3 Quadrilateral tiling 198–223 9 Sequences and functions 9.1 Generating sequences 9.2 Finding rules for sequences 9.3 Using the nth term 9.4 Representing simple functions Algebra We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

5 Contents Page Unit Strand of mathematics 224–234 10 Percentages 10.1 Percentage increases and decreases 10.2 Using a multiplier Number 235–255 11 Graphs 11.1 Functions 11.2 Plotting graphs 11.3 Gradient and intercept 11.4 Interpreting graphs Algebra; Statistics and probability 256 Project 4 Straight line mix-up 257–274 12 Ratio and proportion 12.1 Simplifying ratios 12.2 Sharing in a ratio 12.3 Ratio and direct proportion Number 275–288 13 Probability 13.1 Calculating probabilities 13.2 Experimental and theoretical probabilities Statistics and probability 289 Project 5 High fives 290–330 14 Position and transformation 14.1 Bearings 14.2 The midpoint of a line segment 14.3 Translating 2D shapes 14.4 Reflecting shapes 14.5 Rotating shapes 14.6 Enlarging shapes Statistics and probability 331–351 15 Distance, area and volume 15.1 Converting between miles and kilometres 15.2 The area of a parallelogram and a trapezium 15.3 Calculating the volume of triangular prisms 15.4 Calculating the surface area of triangular prisms and pyramids Geometry and measure 352 Project 6 Biggest cuboid 353–387 16 Interpreting and discussing results 16.1 Interpreting and drawing frequency diagrams 16.2 Time series graphs 16.3 Stem-and-leaf diagrams 16.4 Pie charts 16.5 Representing data 16.6 Using statistics Statistics and probability 388–394 Glossary and Index We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

How to use this book 6 How to use this book In this book you will find lots of different features to help your learning. Questions to find out what you know already. What you will learn in the unit. Important words to learn. Step-by-step examples showing how to solve a problem. These questions will help you develop your skills of thinking and working mathematically. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

How to use this book 7 These investigations, to be carried out with a partner or in a group, will help develop skills of thinking and working mathematically. Questions to help you think about how you learn. This is what you have learned in the unit. Questions that cover what you have learned in the unit. At the end of several units, there is a project for you to carry out, using what you have learned. You might make something or solve a problem. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

8 The authors and publishers acknowledge the following sources of copyright material and are grateful for the permissions granted. While every effort has been made, it has not always been possible to identify the sources of all the material used, or to trace all copyright holders. If any omissions are brought to our notice, we will be happy to include the appropriate acknowledgements on reprinting. Thanks to the following for permission to reproduce images: Cover Photo: ori-artiste/Getty Images PeskyMonkey/Getty Images; pressureUA/Getty Images; Andriy Onufriyenko/ Getty Images; Jonathan Kitchen/Getty Images; Witthaya Prasongsin/Getty Images; ilyast/Getty Images; Bob Langrish/Getty Images; Sean Gladwell/Getty Images; stilllifephotographer/Getty Images; Abstract Aerial Art/Getty Images; Zsschreiner/ Shutterstock; Leon Ritte/Shutterstock; Carl De Souza/Getty Images; krisanapong detraphiphat/Getty Images; ROBERT BROOK/SCIENCE PHOTO LIBRARY/ Getty Images; Roberto Machado Noa/Getty Images; Tetra Images/Getty Images; Wavebreakmedia/Getty Images; Yuji Sakai/Getty Images; Mina De La O/Getty Images; 661031668/Getty Images; Rizky Panuntun/Getty Images; Bim/Getty Images; Sean Gladwell/Getty Images; Cravetiger/Getty Images; Westend61/Getty Images; mammuth/ Getty Images; AtWaG/Getty Images; Nataliia Tymofieieva/Getty Images; Sir Francis Canker Photography/Getty Images; diego_cervo/Getty Images; Photolibrary/Getty Images; fotograzia/Getty Images; JESPER KLAUSEN/SCIENCE PHOTO LIBRARY/ Getty Images; Fernando Trabanco Fotografía/Getty Images; Dimitri Otis/Getty Images; Halfdark/Getty Images; Hollie Fernando/Getty Images; Berkah/Getty Images; traveler1116/Getty Images; Bill Ross/Getty Images CHRISTOPH BURGSTEDT/ SCIENCE PHOTO LIBRARY/Getty Images; Stanzi11/Getty Image; Peter Cade/ Getty Images; Adam Gault/Getty Images; Maskot/Getty Images; fstop123/Getty Images; Arthur Tilley/Getty Images; Adam Smigielski/Getty Images; SAUL LOEB/ Getty Images; MirageC/Getty Images; Boris SV/Getty Images; FotografiaBasica/Getty Images; Xuanyu Han/Getty Images; Geoff Brightling/Getty Images; Ilona Nagy/Getty Images; GeorgiosArt/Getty Images; Paula Daniëlse/Getty Images; Peter Zelei Images/ Getty Images; Universal History Archive /Getty Images; Kenny Williamson/Getty Images; Andriy Onufriyenko/Getty Images; ShaneMyersPhoto/Getty Images; SEAN GLADWELL/Getty Images; Peter Dazeley/Getty Images. Acknowledgements We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

9 1 Integers Getting started 1 a Find all the prime numbers less than 20. b Show that there are two prime numbers between 20 and 30. 2 a Find all the factors of 18. b Find all the 2-digit multiples of 18. c Find the highest common factor of 18 and 12. d Find the lowest common multiple of 18 and 12. 3 Work out a − +6 3 b − − 6 3 c − ×6 3 d − ÷6 3 e 8 10 + − f − − 5 9 − 4 Write whether each of these numbers is a square number, a cube number or both. a 49 b 27 c 1000 d 64 e 121 f 225 5 Find a 100 b 125 3 c 15 12 2 2 − Prime numbers have exactly two factors, 1 and the number itself. Some examples of prime numbers are 7, 31, 83, 239 and 953. The number 39 is the product of two prime numbers (3 and 13). It is quite easy to find these two numbers. The number 2573 is also the product of two prime numbers (31 and 83). It is much harder to find the two numbers in this case. It is easy to multiply two prime numbers together using a calculator or a computer. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

1 Integers 10 It is much harder to carry out the inverse operation – that is, to find the two prime numbers that multiply to a given product. This fact is the basis of a system used to encode messages sent across the internet. The RSA cryptosystem was invented by Ronald Rivest, Adi Shamir and Leonard Adleman in 1977. It uses two large prime numbers with about 150 digits each. These numbers are kept secret, but anybody can use their product, N, which has about 300 digits. If someone sends their credit card number to a website, their computer does a calculation using N to encode their credit card number. The computer that receives the coded number does another calculation to decode it. Anyone who does not know the two factors of N will not be able to do this. Your credit card number is protected. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

11 1.1 Factors, multiples and primes In this section you will … • write a positive integer as a product of prime factors • use prime factors to find a highest common factor (HCF) and a lowest common multiple (LCM). 1.1 Factors, multiples and primes Any integer bigger than 1: • is a prime number, or • can be written as a product of prime numbers. Example: 46=2×23 47 is prime 48=2×2×2×2×3 49=7×7 50=2×5×5 You can use a factor tree to write an integer as a product of its prime factors. This is how to draw a factor tree for 120. 1 Write 120. 2 Draw branches to two numbers that have a product of 120. Do not use 1 as one of the numbers. Here we have chosen 12 and 10. 120 12 10 = × 3 Do the same with 12 and 10. Here 12=3×4 and 10=2×5 4 3, 2 and 5 are prime numbers, so circle them. 5 Draw two more branches from 4. 4=2×2. Circle the 2s. 6 Now all the end numbers are prime, so stop. 7 120 is the product of all the end numbers: 120 = 222 ××× 3 5 × 8 You can check that this is correct using a calculator. You can also write the result like this: 120 2 3 5 3 = × × 23 means 2×2×2 and the small 3 is an index. Now check that 75 3 52 = × You can use products of prime factors to find the HCF and LCM of two numbers. Key words factor tree highest common factor (HCF) index integer lowest common multiple (LCM) prime factor 120 12 10 3 4 2 2 2 5 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

12 1 Integers Exercise 1.1 Worked example 1.1 a Find the LCM of 120 and 75. b Find the HCF of 120 and 75. Answer a Write 120 and 75 as products of their prime factors: 120 = 222 ××× 3 5 × 75 3 = × 5 5 × Look at the prime factors of both numbers. For the LCM, use the larger frequency of each prime factor. • 120 has three 2s and 75 has no 2s. The LCM must have three 2s. • 120 has one 3 and 75 has one 3. The LCM must have one 3. • 120 has one 5 and 75 has two 5s. The LCM must have two 5s. The LCM is 2 2 2 3 5 5 2 35 83 25 600 3 2 ××× × × = × × = × × = b For the HCF use the smaller frequency of each factor: there are no 2s in 75, and there is one 3 and one 5 in both numbers. Multiply these factors. The HCF is 3×5=15 Think like a mathematician 1 The factor tree for 120 in Section 1.1 started with 12×10. a Draw a factor tree for 120 that starts with 6×20. b Compare your answer to part a with a partner’s. Are your trees the same or different? c Draw some different factor trees for 120. Can you say how many different trees are possible? d Do all factor trees for 120 have the same end points? 120 6 20 2 a Complete this factor tree for 108. b Draw a different factor tree for 108. c Write 108 as a product of its prime factors. d Compare your factor trees and your product of prime factors with a partner’s. Have you drawn the same trees or different ones? Are your trees correct? 108 2 54 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

13 1.1 Factors, multiples and primes 3 a Draw a factor tree for 200 that starts with 10×20. b Write 200 as a product of prime numbers. c Compare your factor tree with a partner’s. Have you drawn the same tree or different ones? Are your trees correct? d How many different factor trees can you draw for 200 that start with 10×20? 4 a Draw a factor tree for 330. b Write 330 as a product of prime numbers. 5 Match each number to a product of prime factors. The first one has been done for you: a and i. a 20 i 2²×5 b 24 ii 2×3×7 c 42 iii 2²×3²×5 d 50 iv 2×5² e 180 v 2³×3 6 Work out the product of each set of prime factors. a 3 5 7 2 × × b 2 5 3 3 × c 2 3 11 2 2 × × d 2 7 4 2 × e 3 172 × 7 Write each of these numbers as a product of prime factors. a 28 b 60 c 72 d 153 e 190 f 275 8 a Copy the table and write each number as a product of prime numbers. Number Product of prime numbers 35 5×7 70 140 280 b Add more rows to the table to continue the pattern. 9 a Write 1001 as a product of prime numbers. b Write 4004 as a product of prime numbers. c Write 6006 as a product of prime numbers. 10 a Use a factor tree to write 132 as a product of prime numbers. b Write 150 as a product of prime numbers. c 132×150=19 800. Use this fact to write 19 800 as a product of prime numbers. Tip You can use a factor tree to help you. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

14 1 Integers 11 a Write each of these numbers as a product of prime numbers. i 15 ii 15² iii 28 iv 28² v 36 vi 36² b What do you notice about your answers to i and ii, iii and iv, v and vi? c If 96 2 3 5 = × , show how to find the prime factors of 962 . Will your method work for all numbers? 12 40=2×2×2×5 and 28=2×2×7 Use these facts to find a the HCF of 40 and 28 b the LCM of 40 and 28. 13 450=2×3×3×5×5 and 60=2×2×3×5 Use these facts to find a the HCF of 450 and 60 b the LCM of 450 and 60. 14 180=2²×3²×5 and 54=2×3³ Use these facts to find a the HCF of 180 and 54 b the LCM of 180 and 54. 15 a Write 45 as a product of prime numbers. b Write 75 as a product of prime numbers. c Find the LCM of 45 and 75. d Find the HCF of 45 and 75. 16 a Draw factor trees to find the LCM of 90 and 140. b Compare your answer with a partner’s. Did you draw the same factor trees? Have you both got the same answer? 17 a Write 396 as a product of prime numbers. b Write 168 as a product of prime numbers. c Find the HCF of 396 and 168. d Find the LCM of 396 and 168. 18 a Find the HCF of 34 and 58. b Find the LCM of 34 and 58. 19 Show that the HCF of 63 and 110 is 1. 20 37 and 47 are prime numbers. a What is the HCF of 37 and 47? b What is the LCM of 37 and 47? c Write a rule for finding the HCF and LCM of two prime numbers. d Compare your answer to part c with a partner’s answer. Check your rules by finding the HCF and LCM of 39 and 83. Tip Use a calculator to help you. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

15 1.2 Multiplying and dividing integers In this exercise you have: • used factor trees to write an integer as a product of prime factors • found the HCF of two integers by first writing each one as a product of prime numbers • found the LCM of two integers by first writing each one as a product of prime numbers. a Which questions have you found the easiest? Explain why. b Which questions have you found the hardest? Explain why. Summary checklist I can write an integer as a product of prime numbers. I can find the HCF and LCM of two integers by first writing each one as a product of prime numbers. You can add and subtract any two integers. For example: 2+−4=−2 −2+−4=−6 −2 − 4=−6 −2 − −4=2 You can also multiply and divide a negative integer by a positive one. For example: 2×−9=−18 −6×3=−18 −18÷3=−6 20÷−5=−4 In this section you will investigate how to multiply or divide any two integers. You will use number patterns to do this. 1.2 Multiplying and dividing integers In this section you will … • multiply and divide integers, in particular when both are negative • understand that brackets, indices and operations follow a particular order. Key words brackets conjecture inverse investigate We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

16 1 Integers Exercise 1.2 Worked example 1.2 Look at this sequence of subtractions. 3 − = 6 3− 3 − = 4 1− 3 2 − = 3 0 − = 3 2 − − = 3 4 − − = a Copy the sequence and fill in the missing answers. b Write the next three lines in the sequence. c Describe any patterns in the sequence. Answer a 3 − = 2 1 3 − = 0 3 3 − −2 5 = 3 − −4 7 = b 3 − −6 9 = 3 − −8 1 = 1 3 10 13 − − = c The first number, 3, does not change. The number being subtracted decreases by 2 each time. The answer increases by 2 each time. A sequence is a set of numbers or expressions made and written in order, according to some pattern. Think like a mathematician 1 Here is the start of a sequence of multiplications. − × 3 4 = −12 − × 3 3 = − × 3 2 = a Copy the sequence and write six more terms. Use a pattern to fill in the answers. b Describe the patterns in the sequence. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

17 1.2 Multiplying and dividing integers 2 Work out these multiplications. a 5 2 × − b − × 5 2 c − × 5 2 − d − × 2 5 − 3 Work out these multiplications. a − × 6 4 − b − × 7 7 − c −10×−6 d −8×−11 Continued c Here is the start of another sequence of multiplications. − × 5 4 = − × 5 3 = − × 5 2 = Copy the sequence and write six more terms. Describe any patterns in the sequence. d In the sequences in a and c, you have some products of two negative integers. What can you say about the product of two negative integers? e Make up a sequence of your own like the ones in a and c. f Share your answers to parts d and e with a partner. Are your partner’s sequences correct? 4 Copy and complete this multiplication table. × −5 3 −8 4 −3 −9 −6 30 5 Work out a (3+5)×−4 b (−3+−5)×−6 c −4×(5 − 8) d −6×(−2 − −7) 6 Round these numbers to the nearest whole number to estimate the answer. a 3.9×−6.8 b −11.2×2.95 c (−6.1)2 d (−4.88)2 7 a Put these multiplications into groups based on the answers. 3×−4 −6×−2 12×1 −4×−3 2×−6 −12×−1 b Find one more product to put in each group. Tip Do the calculation in brackets first. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

18 1 Integers 8 These are multiplication pyramids. a –8 2 –4 –3 b –3 5 –1 c –4 –5 –2 Each number is the product of the two numbers below it. For example, in a, 2×−4=−8 Copy and complete the multiplication pyramids. 9 a Draw a multiplication pyramid like those in Question 8, with the integers −2, 3 and −5 in the bottom row, in that order. Complete your pyramid. If you change the order of the bottom numbers, the number at the top of the pyramid is the same. b Is Zara correct? Test her idea by changing the order of the numbers in the bottom row of your pyramid. 10 Find the missing numbers in these multiplications. a −3× =−12 b −5× =45 c ×−6=24 d ×−10=80 Think like a mathematician 11 A multiplication can be written as a division. For example, 5×8=40 can be written as 40÷8=5 or 40÷5=8 a Here is a multiplication: −4×6=−24 Write it as a division in two different ways. b Write a multiplication of a positive integer and a negative integer. Then write it as a division in two different ways. c Here is a multiplication: −7×−2=14 Write it as a division in two different ways. d Write a multiplication of two negative integers. Then write it as a division in two different ways. Tip A conjecture is a possible value based on what you know. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

19 1.2 Multiplying and dividing integers 12 Work out these divisions. a 18÷−6 b −28÷−4 c 30÷−6 d −30÷−10 e 42÷−6 f −24÷−4 g 60÷−5 h −25÷−5 13 Here are three multiplication pyramids. a 6 5 –1 b 12 –2 –8 c –20 –200 –4 Copy and complete each pyramid. 14 Work out a (3×−4)÷−2 b (2 − 20)÷−3 c (−3+15)÷−4 d 24÷(2×−4) 15 Find the value of x. a x÷−4=8 b x÷−3=−15 c 16÷x=−2 d −15÷x=3 16 Round these numbers to the nearest whole number to estimate the answer. a −8.75÷2.8 b 18.1÷−5.9 c −28.2÷−3.8 d −35.2÷−6.9 17 Round these numbers to the nearest 10 to estimate the answer. a −48×−29 b −18.1×61.5 c −71.4÷−11.8 d −99.4÷19 Continued e Can you make a conjecture about the answer when you divide an integer by a negative integer? Test your conjecture. f Compare your answer with a partner’s answers. Have you made the same conjectures? Tip Remember, division is the inverse of multiplication so you will divide as you work down the pyramid. Summary checklist I can multiply two negative integers. I can divide any integer by a negative integer. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

20 1 Integers In this section you will … • find the squares of positive and negative integers and their corresponding square roots • find the cubes of positive and negative integers and their corresponding cube roots • learn to recognise natural numbers, integers and rational numbers. 1.3 Square roots and cube roots 52=25 This means that the square root of 25 is 5. This can be written as 25 5 = . This is the only answer in the set of natural numbers. However (−5)2=−5×−5=25 This means that the integer −5 is also a square root of 25. Every positive integer has two square roots, one positive and one negative. 5 is the positive square root of 25 and −5 is the negative square root. No negative number has a square root. For example, the integer −25 has no square root because the equation x2=−25 has no solution. 53=125 This means that the cube root of 125 is 5. This can be written as 125 5 3 = . You might think −5 is also a cube root of 125. However (−5)3=−5×−5×−5=(−5×−5)×−5=25×−5=−125 So − = 125 5− 3 Every number, positive or negative or zero, has only one cube root. Key words cube root natural numbers rational numbers square root Tip The natural numbers are the counting numbers and zero. Worked example 1.3 Solve each equation. a x2=64 b x3=64 c x3+64=0 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

21 1.3 Square roots and cube roots Exercise 1.3 1 Work out a 72 b (−7)2 c 73 d (−7)3 2 Find a 125 3 b −27 3 c −1 3 d −8 3 3 Solve these equations. a x2=100 b x2=144 c x2=1 d x2=0 e x2+9=0 4 Solve these equations. a x3=216 b x3+27=0 c x3+1=0 d x3+125=0 5 272=93=729 Use this fact to find a 729 b −729 c 729 3 d −729 3 6 a A calculator shows that 8 8 0 2 2 − −( ) = Explain why this is correct. b Find the value of 4 4 3 3 3 3 − −( ) . Show your working. 7 The square of an integer is 100. What can you say about the cube of the integer? 8 The integer 1521=32×132 Use this fact to a find 1521 b solve the equation x2=1521 9 a How is −52 different from (−5)2 ? b What is the difference between −53 and (−5)3 ? Continued Answer a 64 has two square roots. One is 64 8 = and the other is − = 64 8− So the equation has two solutions: x=8 or x=−8 b 64 4 3 = . This means 43=4×4×4=64 and so x=4 c If x3+64=0 then x3=−64. So x = −64 4 = − 3 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

22 1 Integers 12 a Copy and complete this table. x x − 1 x3 − 1 x2+x+1 2 1 7 3 2 13 4 5 b What pattern can you see in your answers? c Add another row to see if the pattern is still the same. d Add three rows where x is a negative integer. Is the pattern still the same if x is a negative integer? e Compare your answers with a partner’s. 10 a Show that 32+42=52 b Are these statements true or false? Give a reason for your answer each time. i (−3)2+(−4)2=(−5)2 ii (−13)2=122+(−5)2 iii 82=−102 − 62 c Show your work to a partner. Do they find your explanation clear? Think like a mathematician 11 a Here is an equation: x2+x=6 i Show that x=2 is a solution of the equation. ii Show that x=−3 is a solution of the equation. b Here is another equation: x2+x=12 i Show that x=3 is a solution of the equation. ii Find a second solution to the equation. c Find two solutions to this equation: x2+x=20 d What patterns can you see in the answers to a, b and c? Find some more equations like this and write down the solutions. e Compare your answers with a partner’s. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

23 1.3 Square roots and cube roots 13 Any number that can be written as a fraction is a rational number. Examples are 7 3 4 , −12 18 25, 6, 1 15, −2 9 10 Here is a list of six numbers: 5 − 1 5 −500 16 −4.8 99 1 2 Write a all the integers in the list b all the natural numbers in the list c all the rational numbers in the list. 14 This Venn diagram shows the relationship between natural numbers and integers. N stands for natural numbers and I for integers. a Copy the Venn diagram. b Write each of these numbers in the correct part of the diagram. 1 −3 7 −12 41 −100 2 1 2 c Add another circle to your Venn diagram to show rational numbers. d Add these numbers to your Venn diagram. −8 3 7 3 5 0 6.3 −10 3 e Give your diagram to a partner to check. N I Tip Remember, all integers are included in the rational numbers. Tip Integers and fractions are included in the set of rational numbers. Summary checklist I can find and recognise square numbers and their two corresponding square roots. I can find and recognise positive and negative cube numbers and their cube roots. I can recognise natural numbers, integers and rational numbers. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

24 1 Integers In this section you will investigate numbers written as powers. Look at these powers of 5 n 0 1 2 3 4 5 5n 25 125 625 3125 So 53=5×5×5=125 and 54=5×5×5×5=625 and so on. As you move to the right the numbers in the bottom row multiply by 5. As you move to the left the numbers in the bottom row divide by 5. 3125÷5=625, 625÷5=125, 125÷5=25 If you continue to divide by 5, 25÷5=5 so 51=5 There is another number missing in the table. What is 50 ? Divide by 5 again: 50=51÷5=5÷5=1 So 50=1 If n is any positive integer then n0=1. 1.4 Indices In this section you will … • use positive and zero indices to represent numbers and in multiplication and division. Key words generalise power Worked example 1.4 a Show that 7 343 3 = b Work out i 74 ii 70 Answer a 7 777 49 7 343 3 = × ×= ×= b i 7 7 7 343 7 2401 4 3 = × = × = ii 7 1 0 = We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

25 1.4 Indices Exercise 1.4 1 Copy and complete this list of powers of 2. Power 20 21 22 23 24 25 26 27 28 29 210 Number 1 2 8 64 512 2 Copy and complete this list of powers of 3. Power 30 31 32 33 34 35 36 37 38 Number 3 27 2187 Think like a mathematician 3 Look at this multiplication: 4×16=64 You can write all the numbers as powers of 2: 22×24=26 a Write each of these multiplications as powers of 2. i 8×4=32 ii 16×8=128 iii 4×32=128 iv 2×128=256 v 16×32=512 b Can you see a pattern in your answers? Make a conjecture about multiplying powers of 2. Test your conjecture on some more multiplications of your own. c Make a conjecture about multiplying powers of 3. Use some examples to test your conjecture. d Generalise your results so far. 4 Write the answers to these calculations as powers of 6. a 6 6 2 3 × b 6 6 4 × c 6 6 5 2 × d 6 6 3 3 × 5 Write the answers to these calculations in index form. a 10 10 3 2 × b 20 20 5× c 15 15 3 3 × d 5 5 5 3 × 6 a 38=6561 Use this fact to find 39 and show your method. b 56=15 625 Use this fact to find 57 and show your method. 7 Find the missing power. a 33×3 =35 b 93×9 =98 c 124×12 =126 d 15 ×153=1510 Tip ‘Generalising’ means using a set of results to come up with a general rule. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

26 1 Integers 8 Read what Sofia says. Is Sofia correct? Give a reason for your answer. 9 A million is 106 . A billion is 1000 million. Write as a power of 10 a one billion b 1000 billion 10 Write in index form a 22×23×2 b 33×34×32 c 5×53×53 d 103×102×104 11 a (32 )3=32×32×32 Write (32 )3 as a single power of 3. b Write in index form i (23 )2 ii (53 )2 iii (42 )3 iv (152 )4 v (104 )3 c N is a positive integer. Write in index form i ( ) N2 3 ii ( ) N4 2 iii ( ) N5 3 d Can you generalise the results of part c? Think like a mathematician 12 Here is a division: 32÷4=8 You can write this using indices: 25÷22=23 a Write each of these divisions using indices. All the numbers are powers of 2 or 3. i 64 ÷ = 4 16 ii 81÷ = 3 27 iii 512 16 32 ÷ = iv 729 ÷ = 9 81 v 9 9 ÷ = 1 b Write some similar divisions using powers of 5. c Can you generalise your results from a and b? Check with some powers of other positive integers. d Compare your results with a partner’s. 42 is equal to 24 and 43 is equal to 34 13 Write the answers to these calculations in index form. a 2 2 7 5 ÷ b 10 10 6 3 ÷ c 15 15 8 6 ÷ d 8 8 10 9 ÷ e 2 2 15 11 ÷ f 2 2 5 5 ÷ 14 Write the answers to these calculations in index form. a 9 9 5 2 × b 9 9 5 2 ÷ c (95 )2 d 5 5 5 4 × e 12 12 8 3 ÷ f (73 )3 g ( )100 4 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

1.4 Indices Summary checklist I can use index notation for positive integers where the index is a positive integer or zero. I can multiply and divide numbers written as powers of a positive integer. 15 Read what Zara says. I think that 5 5 2 3 3 2 ( ) = ( ) a Is Zara correct? Give a reason for your answer. b Is a similar result true for other indices? 16 15=3×5 Use this fact to write as a product of prime factors a 152 b 153 c 155 d 158 17 a Write 5 5 6 4 ÷ as a power of 5. b Write 5 5 6 6 ÷ as a power of 5. c Is it possible to write 5 5 4 6 ÷ as a power of 5? 27 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

28 1 Integers Check your progress 1 a Draw a factor tree for 350. b Write 350 as a product of prime factors. c Write 112 as a product of prime factors. d Find the HCF of 350 and 112. e Find the LCM of 350 and 112. 2 Copy and complete this multiplication table. × −6 −5 7 −10 3 −18 −7 3 Are these calculations correct? If not, correct them. a ( ) − = 5 2− 5 2 b − ×9 1− =1 9− 9 c 45 ÷ −9 6 = − d ( ) − = 10 −1000 3 4 Work out a 40 5 ÷ − b − ÷ 36 6− c 100 ÷ ( ) 2 7 – d ( ) 12 18 3 − − ÷ − 5 Solve these equations. a x2 = 36 b x2 + = 16 0 c x3 = 8 d x3 + = 27 0 6 Work out a ( ) − − 5 4 ( ) − 2 2 b 64 64 3 3 + − 7 Here is an expression: x x 3 2 + Find the value of the expression when a x=3 b x=−3 8 Write as a single power of 8 a 82×83 b 86÷82 c 1 d (83 )3 9 a Write 46 as a power of 2. b Write 94 as a power of 3. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

Getting started 1 Alex thinks of a number, n. Write an expression for the number Alex gets when a she multiplies the number by 2 b she adds 5 to the number. 2 Work out the value of p−q when p=15 and q=3 3 Simplify these expressions by collecting like terms. a 3c+4c+9d−2d b 4xy+7yz−2xy+zy 4 Expand the brackets. a 4(x+3) b 6(2−3y) 5 Solve these equations. a n+12=15 b m−7=2 c 3p=27 d 2r+7=19 6 Write the inequality shown by this number line. 654321 2 Expressions, formulae and equations 29 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

30 2 Expressions, formulae and equations A formula is a set of instructions for working something out. It is a rule written using letters or words. The plural of formula is formulae. People use formulae in everyday life to work out all sorts of things. An employer may use a formula to work out how much to pay the people who work for them. For example, they could use the formula P=R×H, where P is the pay, R is the amount paid per hour and H is the number of hours worked. Doctors may use a formula to assess a person’s health. For example, they could use a formula to find the person’s body mass index (BMI). This formula is: BMI mass height2 = , where the person’s mass is measured in kilograms and their height is measured in metres. If a person’s BMI is too high or too low, the doctor may ask them to lose or put on weight, to make them healthier. 2.1 Constructing expressions In this section you will … • use letters to represent numbers • use the correct order of operations in algebraic expressions • use words or letters to represent a situation. Key words coefficient constant equivalent expression linear expression term unknown variable You can write an algebraic expression by using a letter to represent an unknown number. In the expression 3n+8 there are two terms. 3n is one term. The other term is 8. The letter n is called the variable, because it can have different values. The coefficient of n is 3, because it is the number that multiplies the variable. The number 8 is called a constant. Example: Let n represent a mystery number. You write the number that is 5 more than the mystery number as n+5 or 5+n. You write the number that is three times the mystery number as 3×n or simply 3n. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

31 2.1 Constructing expressions Tip n+5 is the same as 1×n+5. You write the mystery number multiplied by itself as n×n or simply n2 . n+5 and 3n are called linear expressions because the variable is only multiplied by a number. n2 is not a linear expression because the variable is multiplied by itself. Worked example 2.1 Tyler thinks of a number, x. Write an expression for the number Tyler gets when he a doubles the number and subtracts 3 b divides the number by 3 and adds 2 c adds 2 to the number, then multiplies by 4. Answer a 2x−3 Multiply x by 2, then subtract 3. Write 2×x as 2x. b x 3 + 2 Divide x by 3, then add 2. Write x÷3 as x 3 . c 4(x+2) Add 2 to x, then multiply the answer by 4. Write x+2 in brackets to show this must be done before multiplying by 4. Exercise 2.1 1 Copy and complete these sentences. Use the words from the cloud. In the ................... 4x+9, x is a .................... 4x and 9 are ................... of the expression. 4 is the ................... of x. 9 is a .................... The expression is not equal to anything so cannot be .................... 2 a Tanesha has a box that contains x DVDs. Choose the correct expression from the cloud that shows the total number of DVDs she has in the box when i she takes 2 out ii she puts in 2 more iii she takes out half of the DVDs iv she doubles the number of DVDs in the box. b Tanesha starts with 12 DVDs in the box. Work out how many she will have for Question 2a, parts i to iv. solved coefficient terms variable constant expression x 2 x+2 x−2 2x We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

32 2 Expressions, formulae and equations 6 Kia thinks of a number, x. Write an expression for the number Kia gets when she: a divides the number by 3, then adds 1 b adds 1 to the number, then divides by 3 c subtracts 1 from the number, then divides by 3 d divides the number by 3, then subtracts 1. Think like a mathematician 5 In pairs or in a small group, discuss. Sofia and Zara discuss what to write for this problem. ‘I think of a number, n. I halve the number then add 4.’ I think the expression is n 2 + 4 I think the expression is n + 4 2 What do you think? Make a conjecture and convince the other members of your group. Tip Remember the order of operations: Brackets, Indices, Division, Multiplication, Addition, Subtraction 3 a Jake thinks of a number, n. Write an expression for the number Jake gets when he: i multiplies the number by 6, then adds 1 ii divides the number by 4, then adds 5 iii multiplies the number by 2, then subtracts 3 iv divides the number by 10, then subtracts 7. b Jake thinks of the number 20. Work out the numbers he gets in Question 3a, parts i to iv. 4 Match each description with the correct expression. The first one has been done for you: a and iv. a Multiply n by 5 and subtract 4 b Add 4 and n, then multiply by 5 c Multiply n by 5 and add 4 d Add 5 and n, then multiply by 4 e Subtract 4 from n, then multiply by 5 f Subtract 5 from n, then multiply by 4 i 5(n+4) ii 4(n+5) iii 4(n−5) iv 5n−4 v 5n+4 vi 5(4−n) vii 5(n−4) Write a description for the expression that has not been matched. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

33 2.1 Constructing expressions 8 a Sort these cards into groups of equivalent expressions. A 3 4 × x B x + 3 4 C 4 3 x D 4 3 × x E 3 4 × x F 3 4 x G 3 4 + x H 3 4 x I 4 3 x J 3 4 + x b Which card is in a group on its own? 9 This is part of Pedro’s classwork. Are Pedro’s answers correct? If not, write the correct answers for him. Think like a mathematician 7 In pairs or in a small group, discuss. Sofia, Zara and Arun discuss what to write for this problem. ‘I think of a number, n. I divide by 3, then multiply by 2.’ What do you think? Make a conjecture and convince the other members of your group. I think the expression is n 3 × 2 I think the expression is 2 3 n. If you divide by 3 then times by 2, you are finding 2 3 of the number, so you can write 2 3 n Question Write an expression for these. a one-third of x add 4 b 5 subtract two-fifths of y Answers a x 3 +4 b 2y 5 −5 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

34 2 Expressions, formulae and equations 10 a Write an expression for each description. i one-half of x add 8 ii three-quarters of x subtract 12 iii 7 add four-fifths of x iv 20 subtract five-ninths of x b Describe each expression in words. i ii 5 7 4 x − iii 8 2 3 − x iv 3 7 8 + x 11 This is part of Maya’s homework. Question Write an expression for a the perimeter of this rectangle b the area of this rectangle. Answers a perimeter= 1 3a+ 1 3a+4b+4b= 2 3a+8bcm b area= 1 3a × 4b= 4 3abcm2 Use Maya’s method to write an expression for the perimeter and area of each of these rectangles. Simplify each expression. a 6b cm a cm 1 2 d cm 3 5 7c cm x 6 + 2 a cm 1 3 4bcm b Activity 2.1 Work with a partner. Take it in turns to say ‘Write an expression for …’ and give a description like those in Question 10a. For example, ‘Write an expression for two-thirds of x add 9.’ Your partner must write the expression correctly. Check their expression. If it is correct, they score 1 point. Write five expressions each, then check the scores! We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

35 2.1 Constructing expressions 12 The shortest side of a triangle is y cm. The second side is 3 cm longer than the shortest side. The third side is twice as long as the second side. Write an expression, in its simplest form, for the perimeter of the triangle. 13 The price of one bag of cement is $c. The price of one bag of gravel is $g. The price of one bag of sand is $s. Write an expression for the total cost of a one bag of cement and three bags of sand b three bags of cement, four bags of gravel and six bags of sand. 14 The price of one kilogram of apples is $a. The price of one kilogram of bananas is $b. The price of one kilogram of carrots is $c. Write an expression for the total cost of a one kilogram of apples and half a kilogram of bananas b two kilograms of bananas and three-quarters of a kilogram of carrots c three kilograms of apples, a quarter of a kilogram of bananas and four-fifths of a kilogram of carrots. Tip Start by writing expressions for the second and third sides. 15 Brad thinks of a number, y. Choose the correct expression from the cloud for when Brad a adds 5 to one-half of y, then multiplies by 6 b adds 6 to one-fifth of y, then multiplies by 2 c adds 2 to five-sixths of y, then multiplies by 6 d adds 5 to two-fifths of y, then multiplies by 6. 6 2 5 6 y +       6 5 2 y +       6 5 2 5 y +       2 6 5 y +       We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

2 Expressions, formulae and equations 36 Summary checklist I can use letters to represent unknown numbers. I can use the correct order of operations in algebraic expressions. I can use words or letters to represent situations. Which statement best describes how you found the questions in this exercise? A I found the questions very difficult to answer. B I found the questions difficult to answer. C I answered the questions but I had to think carefully. D I found the questions easy to answer. What can you do to improve your knowledge and understanding of this topic? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

37 2.2 Using expressions and formulae A formula is a mathematical rule that shows the relationship between two or more quantities (variables). It is a rule that can be written in letters or words. The plural of formula is formulae. You can write, or derive, your own formulae to solve problems. An example of a formula is F=ma. In this formula, F is the subject of the formula. The variable F is written on its own on the left hand side of the formula. You may need to rearrange a formula to make a different variable the subject. This is called changing the subject of the formula. For example, if you know the values of F and a, and you want to find the value of m, you will rearrange the equation like this: When you substitute numbers into formulae and expressions, remember the order of operations. Brackets and indices must be worked out before divisions and multiplications. Additions and subtractions are always worked out last. 2.2 Using expressions and formulae In this section you will … • use the correct order of operations in algebraic expressions • represent a situation either in words or as a formula • change the subject of a formula. Key words changing the subject derive formula formulae inverse operation solve subject of a formula substitute Tip Examples of indices are 22, 52, 43 and 73. F F a F a ma m m = = = Worked example 2.2 a Work out the value of the expression 2x+4y when x=5 and y=−2. b Work out the value of the expression 3 4 2 x + when x=10. c Write a formula for the number of hours (h) in any number of days (d), using i words ii letters. d Use the formula in part c to work out the number of hours in 7 days. e Rearrange the formula in part cii to make d the subject. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

38 2 Expressions, formulae and equations Continued Answer a 2×5+4×−2 = 10+−8 = 10−8=2 Substitute x=5 and y=−2 into the expression. Work out 2×5 and 4×−2. Adding −8 is the same as subtracting 8. b 3 10 4 2 × + = 3×100+4 = 300+4=304 Substitute x=10 into the expression. Work out 102 first. Work out the multiplication before the addition. c i hours=24×days ii h=24d There are 24 hours in every day. Use h for hours and d for days. d h=24×7=168 Substitute d=7 into the formula. e h=24d h 24 =d d= h 24 The formula is h=24×d Use the inverse operation to make d the subject by dividing by 24. Rewrite the formula with the subject, d, on the left hand side. Exercise 2.2 1 Copy and complete the working to find the value of each expression. a p+5 when p=−3 p+5=−3+5= b q−6 when q=4 q−6=4−6= c 6h when h=−3 6h=6×−3= d j 4 when j=−20 j 4 20 4 = = − e a+b when a=6 and b=−3 a+b=6+−3=6−3= f c−d when c=25 and d=32 c−d=25−32= 2 Work out the value of each expression. a 8m−5 when m=−2 b 3z+v when z=8 and v=−20 c 2x+3y when x=4 and y=5 d 20−3n when n=9 e u 2 − 5 when u=4 f p q 5 2 + when p=30 and q=−8 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

39 2.2 Using expressions and formulae 3 Work out the value of each expression. a x2 + 5 when x=4 b 10 − y2 when y=5 c g2+h2 when g=3 and h=6 d m2−n2 when m=7 and n=8 e 4k2 when k=2 f 3 3 r when r=1 g 2y3 when y=3 h x3 − 5 when x=2 i 20−w3 when w=4 j y2 2 when y=4 Tip Remember that r3 means r×r×r. 4 This is part of Dakarai’s homework. He has made a mistake in his working. Question Work out the value of x2 – 8 when x=–3. Answer x2 – 8=(–3)2 – 8 =–3 × –3 – 8 =–9 – 8 =–17 a Explain the mistake he has made. b Work out the correct answer. c Work out the value of y2 + 4 when y=−5. 5 This is part of Oditi’s homework. She has made a mistake in her working. Question Work out the value of 5x3 when x=–2. Answer 5x3=5 × (–2)3 = (–10)3 =–10 × –10 × –10 =–1000 a Explain the mistake she has made. b Work out the correct answer. c Work out the value of 2y3 when y=−3. 6 a Write a formula for the number of months in any number of years, in i words ii letters. b Use your formula in part aii to work out the number of months in 8 years. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

40 2 Expressions, formulae and equations 7 This is how a taxi company works out the cost of a journey for a customer: There is a fixed charge of $6 plus $2 per kilometre. a Write a formula for the cost of a journey, in i words ii letters. b Use your formula in part aii to work out the cost of a journey of 35 km. 8 Use the formula v=u+10t to work out the value of v when a u=5 and t=12 b u=8 and t=15 c u=0 and t=20. 9 Use the formula F=ma to work out F when a m=6 and a=2 b m=18 and a=3 c m=8 and a=−4. 10 The height of a horse is measured in hands (H) and inches (I). This formula is used to work out the height of a horse in centimetres (C). C=2.5(4H+I) where: C is the number of centimetres H is the number of hands I is the number of inches. Sasha has a horse with a height of 16 hands and 1 inch. She uses the formula to work out the height of her horse, in centimetres. C=2.5(4 × 16+1) = 2.5(64+1) = 2.5 × 65 = 162.5 cm Work out the height, in centimetres, of a horse with height a 14 hands and 2 inches b 15 hands and 3 inches c 13 hands and 1 inch d 17 hands and 2 inches e 16 hands f 12 hands. Tip 16 hands exactly means 16 hands and 0 inches. Would it matter if the formula used the letters D, E and F instead of C, H and I? Do the letters help you to understand a formula? Explain your answer. 11 Use the formula C=πd to a estimate the value of C when d=19 m b calculate the value of C when d=19 m. Give your answer to one decimal place. Tip Remember that π is approximately 3.14. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

41 2.2 Using expressions and formulae 12 Xavier uses this formula to work out the volume of a triangular prism. V bhl = 2 where: V is the volume; b is the base; h is the height; l is the length. Xavier compares two prisms. Prism A has a base of 8 cm, height of 5 cm and length of 18 cm. Prism B has a base of 9 cm, height of 14 cm and length of 6 cm. Xavier works out that Prism A has the larger volume by 12 cm3 . Is Xavier correct? Explain your answer. Tip Remember that bhl means b×h×l. Think like a mathematician 13 Work with a partner to answer this question. Discuss which answers are correct. Identify the mistakes that have led to the incorrect answers. Make x the subject of each formula. Write which answer is correct, A, B or C. a y=x+9 A x=y+9 B x=y−9 C x=9−y b y=mx A x=my B x m y = C x y m = c y=x−c A x=y+c B x=y−c C x=c−y d y x k = A x y k = B x=ky C x k y = e y=7x−3 A x y = + 3 7 B x y = + 7 3 C x y = − 3 7 Tip To answer this question, you will need to critique the given answers and improve them. 14 a Use the formula T=mg to work out the value of T when m=4.5 and g=10. b Rearrange the formula T=mg to make m the subject. c Use your formula to work out the value of m when T=320 and g=10. 15 a Use the formula h=k−d to work out the value of h when k=72 and d=37. b Rearrange the formula h=k−d to make k the subject. c Use your formula to work out the value of k when h=0.42 and d=1.83 Tip Use inverse operations on T=m×g to make the formula m= We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

42 2 Expressions, formulae and equations 16 a Use the formula f w p = to work out the value of f when w=60 and p=12. b Rearrange the formula f w p = to make w the subject. c Use your formula to work out the value of w when f =0.25 and p=52. 17 Polly and Theo use different methods to work out the answer to a question. This is what they write. Question Use the formula P=3n+b to work out the value of n when P=72 and b=6. Answers Polly Theo Step 1: Make n the subject of the formula. P=3n+b P− b=3n P− b 3 =n n= P− b 3 Step 2: Substitute in the numbers. n= 72− 6 3 = 66 3 =22 Step 1: Substitute in the numbers. P=3n+b 72=3n+6 Step 2: Solve the equation. 72−6=3n 66=3n 66 3 =n 22=n n=22 a Look at Polly and Theo’s methods. Do you understand both methods? Do you think you would be able to use both methods? b Which method do you prefer and why? c Use your preferred method to answer these questions. i Use the formula H=6p−k to work out the value of p when H=40 and k=14. ii Use the formula y=mx+7 to work out the value of m when y=25 and x=3. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

43 2.3 Expanding brackets Summary checklist I can substitute numbers into expressions. I can write formulae. I can understand and use formulae. I can change the subject of formulae. Tip Expanding brackets is sometimes called multiplying out brackets. To expand brackets, multiply each term inside the brackets by the term directly in front of the brackets. 2.3 Expanding brackets In this section you will … • expand brackets. Key words expand brackets Worked example 2.3 a Expand these expressions. i 3(2b+5) ii a(a−3) b Expand and simplify this expression. 4(2x+3x2 )−x(6+x) Answer a i 3(2b+5)=3×2b+3×5 =6b+15 ii a(a−3)=a×a−a×3 =a2−3a Multiply 3 by 2b then multiply 3 by 5. Simplify 3×2b to 6b and simplify 3×5 to 15. Multiply a by a then multiply a by −3. Simplify a×a to a2 and simplify a×−3 to −3a. b 4(2x+3x2 )−x(6+x) = 8x+12x2−6x−x2 = 2x+11x2 Start by multiplying out both brackets. So, 4×2x=8x, 4×3x2=12x2 , −x×6=−6x, −x×x=−x2 Collect like terms: 8x−6x=2x and 12x2−x2=11x2 . We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

44 2 Expressions, formulae and equations Exercise 2.3 1 Copy and complete the working. Expand the brackets. a 3(x+4)=3×x+3×4 b 8(y−2)=8×y−8×2 = 3x+ = 8y− c 9(3q−4)=9×3q−9×4 = − 2 Expand each expression. a 4(x+6) b 7(z−2) c 2(a+8) d 6(3−4e) e 2(2p+3q) f 9(6t−2s) g 7(6xy−2z) h 5(2x+y+4) 3 Copy and complete the working. Expand the brackets. a x(y+3)=x×y+x×3 b y(y−2)=y×y−y×2 = xy+ = y2− c p(3+4p)=p×3+p×4p d q(6q−15)=q×6q−q×15 = +4p2 = − 4 Expand each expression. a y(y+8) b z(2w−1) c m(m−4) d n(2n+5) e n(9−8n) f a(1−3b) g e(2e+7f) h g(3h+7g) i h(2h−5k) j d(3c−5e) Think like a mathematician 5 In pairs or a small group, discuss what Zara and Arun say. I don’t, I get 8dc−14da so one of us must be wrong! When I expand 2d(4c−7a), I get 8cd−14ad Do you agree with Arun? Explain your answer. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

45 2.3 Expanding brackets 6 Jing, Jun and Amira compare the methods they use to expand the bracket 5k(6m−8k). Jing uses the method shown in Question 3. Jun uses a multiplication box. Amira uses multiplication arcs. 5k(6m– 8k) =5k × 6m– 5k × 8k =30km– 40k2 × 6m –8k 5k 30km –40k2 So 5k(6m– 8k)= 30km– 40k2 5k(6m– 8k)=30km– 40k2 So 5k(6m–8k)=30km–40k2 a What do you think about Jing, Jun and Amira’s methods? b Which method do you think is best for expanding brackets correctly? Explain why. c Use your favourite method to expand i 2x(x+3y) ii 3y(5y+6) iii 4b(6b−2a) iv 2f(2f+g−3) 7 Here are some expression cards. Sort the cards into groups of equivalent expressions. A 2 8 6 2 x x ( ) + x B 10 3 2 2 x x ( ) + C 2 12 9 2 x x ( + ) D 2 10 15 2 x x ( ) + E 4 4 3 2 x x ( + ) F 3 6 8 2 x x ( + ) G 5 6 4 3 ( ) x x + H 6 3 4 2 x x ( ) + x I x x 2 (12 16 + ) Think like a mathematician 8 Work with a partner to discuss this question. Look at this expansion. x(2x+5)+3x(2x+4)=2 5 6 12 2 2 x x + + x x + a How would the expansion change if the+changed to −? Here is the expansion again. x(2x+5)+3x(2x+4)=2 5 6 12 2 2 x x + + x x + b How would the expansion change if both the+changed to −? c Copy these expansions and fill in the missing signs (+ or −). Tip For card A, 2x(8x2+6x) =16x3+12x2 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

46 2 Expressions, formulae and equations 9 Expand each expression and simplify by collecting like terms. a x(x+2)+x(x+5) b z(2z+1)+z(4z+5) c u(2u+5)−u(u+3) d w(6w+2x)−2w(2w−9x) 10 This is part of Shen’s homework. He has made a mistake in every question. Question Expand and simplify 1 8(x+5) – 3(2x+7) 2 a(2b+c)+b(3c – 2a) 3 2y(y+5x)+x(3x+4y) Answers 1 8(x+5) – 3(2x+7)=8x+40 – 6x+21=2x+61 2 a(2b+c)+b(3c – 2a)=2ab+ac+3bc – 2ab=ac+3bc=3abc2 3 2y(y+5x)+x(3x+4y)=2y2+10xy+9x2+4xy=9x2+2y2+14xy a Explain what Shen has done wrong. b Work out the correct answers. Activity 2.3 Work with a partner to answer this question. Here are six expressions. A x(5x+2)+3x(4x+1) B y(y2+4)+6y2(y+8) C 7 2 7 1 9 2 p p ( ) + − p p + D 6 18345 2 k k + − ( ) − k E 5 4 2 2 2 2 n n( ) − − n n( ) + F 8m(m+3)−2m(4m−3) a Choose one of the expressions and ask your partner to expand the brackets and simplify the expression. While they are working, you work out the answer too. Mark your partner’s work. If your answers are different, discuss any mistakes that have been made. b Now ask your partner to choose an expression for you. Expand the brackets and simplify the expression. Ask your partner to mark your work. Discuss any mistakes that have been made. c Do this twice each, so four of the expressions have been chosen altogether. Summary checklist I can multiply out a bracket and collect like terms. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

47 2.4 Factorising In this section you will … • use the HCF to factorise an expression. To expand a term with brackets, you multiply each term inside the brackets by the term outside the brackets. When you factorise an expression, you take the highest common factor (HCF) and put it outside the brackets. 4x+12=4(x+3) 2.4 Factorising Key words factorisations factorise highest common factor (HCF) Worked example 2.4 Factorise these expressions. a 2x+10 b 8−12y c 4a+8ab d x x 2 − 5 Answer a 2x+10=2(x+5) The HCF of 2x and 10 is 2, so put 2 outside the brackets. Divide both terms by 2 and put the results inside the brackets. Check the answer by expanding: 2×x=2x and 2×5=10. b 8−12y=4(2−3y) The HCF of 8 and 12y is 4, so put 4 outside the brackets. Divide both terms by 4 and put the results inside the brackets. Check the answer by expanding: 4×2=8 and 4×−3y=−12y. c 4a+8ab=4a(1+2b) The HCF of 4a and 8ab is 4a, so put 4a outside the brackets. Divide both terms by 4a and put the results inside the brackets. Check the answer: 4a×1=4a and 4a×2b=8ab. d x2 −5x=x(x−5) The HCF of x2 and 5x is x, so put x outside the brackets. Divide both terms by x and put the results inside the brackets. Check the answer: x×x=x2 and x×−5=−5x. 4(x+3)=4x+12 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

48 2 Expressions, formulae and equations Exercise 2.4 1 Copy and complete these factorisations. All the numbers you need are in the cloud. a 3x+15=3(x+ ) b 10y−15=5(2y− ) c 14−28x=7( − 4x) d 12−9y=3( −3y) 2 Copy and complete these factorisations. All the numbers you need are in the cloud. a 4x2+5x=x(4x+ ) b 6xy+12y=6y(x+ ) c 7y−7y2=7y( −y) d 21x−12xy=3x( −4y) Think like a mathematician 3 In pairs or a small group, discuss what Marcus and Arun say. When I factorise 6x+18 I get 3(2x+6) I don’t, I get 6(x+3) so one of us must be wrong! Do you agree with Arun? Explain your answer. 2 3 4 5 1 2 5 7 4 Factorise each of these expressions. Each one has a highest common factor of 2. a 2x+4 b 4b−6 c 8+10y d 18−20m 5 Factorise each of these expressions. Each one has a highest common factor of 3. a 18+21p b 3y−18 c 9+15m d 12−27x 6 Factorise each of these expressions. Make sure you use the highest common factor. a 10z+5 b 8a−4 c 14+21x d 18−24z We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE

49 2.4 Factorising 8 Each expression on a yellow card has been factorised to give an expression on a blue card. Match each yellow card with the correct blue card. A 6x2+12x B 6x2+15x C 6x2+9x D 6x2+18x i 3x(2x+5) ii 6x(x+3) iii 6x(x+2) iv 3x(2x+3) Think like a mathematician 7 In pairs or a small group, discuss what Zara and Sofia say. I think the highest common factor of 6x and 9x2 is 3. I think the highest common factor of 6x and 9x2 is 3x. a What do you think? Explain your answer. b What is the highest common factor of i 8y and 4y2 ii 12p2 and 15p iii 4ab and 5a? 9 Factorise each of these expressions. a 3x2+x b 6y2−12y c 3b+9b2 d 12n−15n2 e 18y−9x f 12y+9x g 8xy−4y h 15z+10yz 10 Copy and complete these factorisations. a 2x+6y+8=2(x+3y+ ) b 4y−8+4x=4(y− +x) c 9xy+12y−15=3(3xy+ −5) d 5x2+2x+xy=x(5x+ + ) e 9y−y2−xy=y( − − ) f 3y2−9y+6xy=3y( − + ) We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE