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C a Cambridge IGCSE m b Mathematics rid g Second edition e IG Karen Morrison and Nick Hamshaw C S E M Cambridge IGCSE Physics IGCSE is a resource that has been written a t and developed to provide full support for the IGCSE h e syllabus offered by CIE (0625). The book is in full m a colour and includes a free CDROM containing a t ic Mathematics range of supporting activities. The materials are s S fully endorsed by CIE. e c o t n The author has substantial experience of teaching d e and developing physics resources for a range of d f markets and is a member of the Institute of Physics. itio Core and Extended — Student’s CD‐ROM offers full support for n a independent learning and a dedicated miniwebsite. — Topics are all introduced in terms of their relevance toB liafec ikn Cthoev 2e1rs tC coepntyu tryo. add Coursebook — Example examination questions and worked r answers with commentary. M Cambridge University Press and Cambridge o D r r International Examinations unique partnership is o enables thousands of students to successfully pass n their exams through the provision of comprehensive, high‐quality and endorsed resources. Cambridge University Press works closely with CIE, and experienced authors, to produce high‐quality and endorsed textbooks and software that support Cambridge teachers and encourage Cambridge learners. To find out more about Cambridge International Examinations www.cie.org.uk To find out more about Cambridge University Press visit www.cambridge.org/cie Endorsed by Endorsed by University of Cambridge International Examinations © Cambridge University Press 2012 Karen Morrison and Nick Hamshaw Cambridge IGCSE® Mathematics Core t f and Eaxtended r Coursebook D © Cambridge University Press 2012 IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd ii 33//99//22001122 77::0000::1155 PPMM cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai Cambridge University Press Th e Edinburgh Building, Cambridge CB2 8RU, UK www.cambridge.org Information on this title: www.cambridge.org/9781107606272 © Cambridge University Press 2012 Th is 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. t First published 2012 Printed in the United Kingdom at the University Press, Cambridge f A catalogue record for this publication is available from the British Library a ISBN-13 978-1-107-60627-2 Paperback with CD-ROM Cover image: Seamus Ditmeyer/Alamy r Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in D 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 are correct at the time of fi rst printing but Cambridge University Press does not guarantee the accuracy of such information thereaft er. notice to teachers References to Activities contained in these resources are provided ‘as is’ and information provided is on the understanding that teachers and technicians shall undertake a thorough and appropriate risk assessment before undertaking any of the Activities listed. Cambridge University Press makes no warranties, representations or claims of any kind concerning the Activities. To the extent permitted by law, Cambridge University Press will not be liable for any loss, injury, claim, liability or damage of any kind resulting from the use of the Activities. IGCSE® is the registered trademark of University of Cambridge International Examinations. © Cambridge University Press 2012 IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd iiii 33//99//22001122 77::0000::1188 PPMM Contents Introduction v Acknowledgements vii Unit 1 Chapter 1: Reviewing number concepts Chapter 3: Lines, angles and shapes 1.1 Different types of numbers 2 3.1 Lines and angles 44 1.2 Multiples and factors 3 3.2 Triangles 53 1.3 Prime numbers 6 3.3 Quadrilaterals 57 1.4 Powers and roots 10 3.4 Polygons 59 1.5 Working with directed numbers 13 3.5 Circles 62 1.6 Order of operations 14 3.6 Construction 63 1.7 Rounding numbers 18 t Chapter 4: Collecting, organising and Chapter 2: Making sense of algebra displaying data 2.1 U sing letters to represent 4f.1 Collecting and classifying data 74 unknown values 23 4.2 Organising data 76 2.2 Substitution 25 4.3 Using charts to display data 83 a 2.3 Simplifying expressions 27 2.4 Working with brackets 31 2.5 Indices 33 Unit 2 r Chapter 5: Fractions Chapter 7: Perimeter, area and volume 5.1 Equivalent fractions 99 7.1 Perimeter and area in two-dimensions 129 5.2 Operations on fractions 100 7.2 Three-dimensional objects 141 D 5.3 Percentages 105 7.3 Surface areas and volumes of solids 143 5.4 Standard form 110 5.5 Your calculator and standard form 114 Chapter 8: Introduction to probability 5.6 Estimation 115 8.1 Basic probability 154 8.2 Theoretical probability 154 Chapter 6: Equations and transforming formulae 8.3 T he probability that an event does not 6.1 F urther expansions of brackets 119 happen 156 6.2 Solving linear equations 121 8.4 Possibility diagrams 158 6.3 Factorising algebraic expressions 123 8.5 C ombining independent and mutually 6.4 Transformation of a formula 124 exclusive events 160 Unit 3 Chapter 9: Sequences and sets 11.3 Understanding similar shapes 216 9.1 Sequences 165 11.4 Understanding congruence 224 9.2 Rational and irrational numbers 170 Chapter 12: Averages and measures of spread 9.3 Sets 172 12.1 Different types of average 231 Chapter 10: Straight lines and quadratic equations 12.2 Making comparisons using averages 10.1 Straight lines 184 and ranges 233 10.2 Quadratic expressions 198 12.3 Calculating averages and ranges for Chapter 11: Pythagoras’ theorem and frequency data 235 similar shapes 12.4 Calculating averages and ranges for grouped 11.1 Pythagoras’ theorem 207 continuous data 237 11.2 Understanding similar triangles 211 12.5 Percentiles and quartiles 240 Examination practice: structured question for Unit 1-3 245 © Cambridge University Press 2012 Contents iii IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd SSeecc11::iiiiii 33//99//22001122 77::0000::1188 PPMM Unit 4 Chapter 13: Understanding measurement Chapter 15: Scale drawings, bearings and 13.1 Understanding units 251 trigonometry 13.2 Time 253 15.1 Scale Drawings 300 13.3 Upper and lower bounds 257 15.2 Bearings 302 13.4 Conversion graphs 262 15.3 Understanding the tangent, cosine 13.5 More money 264 and sine ratios 303 15.4 Solving problems using Chapter 14: Further solving of equations and trigonometry 318 inequalities 15.5 Angles between 90° and 180° 322 14.1 Simultaneous linear equations 268 15.6 The sine and cosine rules 323 14.2 Linear inequalities 275 15.7 Area of a triangle 331 14.3 Regions in a plane 279 15.8 Trigonometry in three-dimensions 334 14.4 Linear programming 284 14.5 Completing the square 286 Chapter 16: Scatter diagrams 14.6 Quadratic formula 287 and correlation t 14.7 Factorising quadratics where the coeffi cient 16.1 Introduction to bivariate data 342 of ‘x2‘s is not 1 289 14.8 Algebraic fractions 291 f Unit 5 a Chapter 17: Managing money Chapter 19: Symmetry and loci 17.1 Earning money 352 19.1 Symmetry in two-dimensions 398 17.2 Borrowing and investing money 357 19.2 Symmetry in three-dimensions 401 17.3 Buying and selling 363 19.3 Symmetry properties of circles 404 r 19.4 Angle relationships in circles 407 Chapter 18: Curved graphs 19.5 Locus 412 18.1 Plotting quadratic graphs (the parabola) 371 Chapter 20: Histograms and frequency distribution 18.2 Plotting reciprocal graphs (the hyperbola) 375 D diagrams 18.3 Using graphs to solve quadratic equations 377 20.1 Histograms 420 18.4 Using graphs to solve simultaneous linear 20.2 Cumulative frequency 426 and non-linear equations 379 18.5 Other non-linear graphs 381 18.6 Finding the gradient of a curve 390 Unit 6 Chapter 21: Ratio, rate and proportion Chapter 23: Transformations and matrices 21.1 Working with ratio 442 23.1 Simple plane transformations 488 21.2 Ratio and scale 446 23.2 Vectors 500 21.3 Rates 450 23.3 Further transformations 512 21.4 Kinematic graphs 452 23.4 Matrices and matrix transformation 517 21.5 Proportion 460 23.5 Matrices and transformations 525 21.6 Direct and inverse proportion in Chapter 24: Probability using tree diagrams algebraic terms 463 24.1 Using tree diagrams to show outcomes 534 21.7 Increasing and decreasing amounts 24.2 Calculating probability from tree diagrams 535 by a given ratio 467 Chapter 22: More equations, formulae and functions 22.1 Setting up equations to solve problems 472 22.2 Using and transforming formulae 474 22.3 Functions and function notation 477 Examination practice: structured question for Unit 4-6 542 Exercise Answers 552 Glossary 615 Index 620 iv Contents © Cambridge University Press 2012 IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd SSeecc11::iivv 33//99//22001122 77::0000::1188 PPMM Introduction Th is highly illustrated coursebook covers the complete Cambridge IGCSE Mathematics (0580) syllabus. Core and Extended material is combined in one book, off ering a one-stop-shop for all students and teachers. Useful hints are included in the margins for students needing more support, leaving the narrative clear and to the point. Th e material required for the Extended course is clearly marked using colour panels and these students are given access to the parts of the Core syllabus they need without having to use an additional book. Th e coursebook has been written with a clear progression from start to fi nish, with some later REWIND chapters requiring knowledge learnt in earlier chapters. Th ere are useful signposts throughout You learned how to plot lines from the coursebook that link the content of the chapters, allowing the individual to follow their own equations in chapter 10. (cid:87) course through the book: where the content in one chapter might require knowledge from a t previous chapter, a comment is included in a ‘Rewind’ box; and where content will be covered in FAST FORWARD more detail later on in the coursebook, a comment is included in a ‘Fast forward’ box. Examples f You will learn much more about of both are included here. sets in chapter 9. For now, just think Worked examples are used throughout to demonstrate each method using typical workings and of a set as a list of numbers or other thought processes. Th ese present tahe methods to the students in a practical and easy-to-follow items that are often placed inside way that minimises the need for lengthy explanations. curly brackets. (cid:88) Th ere is plenty of practice off ered via ‘drill’ exercises throughout each chapter. Th e exercises are progressive questions which allow the student to practise methods that have just been introduced. At the end of each chapter there are ‘Exam-style’ questions and ‘Past paper’ r questions. Th e exam-style questions have been written by the authors in the style of questions on exam papers. Th e past paper questions are real questions taken from past exam papers. Th ese end of chapDter questions typically refl ect the ‘short’ Paper 1 (Core) and Paper 2 (Extended) questions, though you will fi nd some more structured ones in there as well. Th e answers to all of these questions are supplied at the back of the book, allowing self- and/or class- assessment. Students can assess their progress as they go along, choosing to do more or less practise as required. Th e ‘suggested’ progression through the coursebook is for Units 1-3 to be covered in the fi rst year of both courses, and Units 4-6 to be covered in the second year of both courses. On this basis, there is mixed exam practice at the end of Unit 3 and the end of Unit 6. Th is is however, only a suggested structure and the course can be taught in various diff erent ways; the signposting throughout the coursebook means that it can be used alongside any order of teaching. Th e end of Unit questions represent the longer answer ‘structured’ questions of Paper 3 (Core) and Paper 4 (Extended) exam papers and will use a combination of methods from across all relevant chapters. As with the end of chapter questions, these are a mixture of ‘Exam-style’ and ‘Past paper’ questions. Th e answers to these questions are on the Teacher’s resource so that they can be used in classroom tests or for homework, if desired. Th e coursebook also comes with a glossary to provide a defi nition for important / tricky terms. Helpful guides in the margin of the book include: Hints: these are general comments to remind you of important or key information that is useful Remember ‘coeffi cient’ is the to tackle an exercise, or simply useful to know. Th ey oft en provide extra information or support number in the term. in potentially tricky topics. © Cambridge University Press 2012 Introduction v IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd SSeecc22::vv 33//99//22001122 77::0000::1199 PPMM Tip: these are tips that relate to the exam. Th ey cover common pitfalls based on the authors’ Tip experiences of their students, and give you things to be wary of or to remember in order to It is essential that you score marks in the exam. Please note that this advice is not from the University of Cambridge remember to work out International Examinations Syndicate and they bear no responsibility for any such advice given. both unknowns. Every Th e accompanying student CD-ROM at the back of the coursebook includes: pair of simultaneous linear • equations will have a pair A ‘coverage grid’ to map the contents of the syllabus to the topics and chapters in the of solutions. coursebook. • A ‘Calculator support’ chapter. Th is chapter covers the main uses of calculators that students seem to struggle with, and includes some worksheets to provide practice at using your calculator in these situations. • Revision • Core revision worksheets (and answers) provide extra exercises for each chapter of the t book. Th ese worksheets contain only content from the Core syllabus. • Extended revision worksheets (and answers) provide extra exercises for each chapter f of the book. Th ese worksheets contain the same questions as the Core worksheets, in addition to some more challenging questions, and questions to cover content unique to the Extended syllabus. Staudents are encouraged to do some (if not all) of the ‘Core’ questions on these worksheets, as well as the Extended ones (shaded) in order to fully revise the course. If time is limited, you might fi nd it easier to pick two or three ‘Core’ questions to do before moving on to the ‘Extended’ questions. • Quick revision tests – these are interactive questions in the form of multiple choice, drag r and drop, or hide and reveal. Th ey are quick-fi re questions to test yourself in a diff erent medium to pen and paper, and to get you thinking on the spot. Th ey cover the Core contDent, with only a few additional screens being specifi c to the Extended course. Th ere is at least one activity for each chapter. Students are recommended to use the Revision worksheets for a more comprehensive revision exercise. • Worked solutions – these are interactive hide and reveal screens showing worked solutions to some of the end of chapter examination practice questions. Some of these will be ‘Exam-style’ and some will be ‘Past paper’ questions but all will be taken from the end of the chapter. Th ere will be at least one for each chapter. Th e screen includes the question, and the answer, but also includes a series of ‘Clue’ or ‘Tip’ boxes. Th e ‘Clue’ boxes can be clicked on to reveal a clue to help the student if they are struggling with how to approach the question. Th e ‘Tip’ boxes contain tips relating to the exam, just like the ‘Tip’ boxes in the coursebook. Also in the Cambridge IGCSE Mathematics series are two Practice Books – one for Core and one for Extended – to off er students targeted practice. Th ese follow the chapters and topics of the coursebook, including additional exercises for those who want more practice. Th ese too, include ‘Hints’ and ‘Tips’ to help with tricky topics. vi Introduction © Cambridge University Press 2012 IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd SSeecc22::vvii 33//99//22001122 77::0000::1199 PPMM Acknowledgements Th e authors and publishers acknowledge the following sources of copyright material and are grateful for the permissions granted. Past paper examination questions are reproduced by permission of University of Cambridge International Examinations Cover image Seamus Ditmeyer/Alamy; p. 1 © sanderderwilde.com; p. 13 Dmitry Lavruhin/Shutterstock; p. 13 Hadriann/Shutterstock; p. 13 Jason Cox/Shutterstock; p. 13 Ruslan Nabiyev/Shutterstock; p. 46 Ivangott/Shutterstock; p. 47 Claudio Baldini/ Shutterstock; p. 63 akiyoko/Shutterstock; p. 73 INSAGO/Shutterstock; p. 98 © Th e Trustees of the British Museum; p. 119 Wikipedia; p. 128 Francesco Dazzi/Shutterstock; p. 164 North Wind Picture Archive/Alamy; p. 148 Paolo Gianti/Shutterstock; pp. 153, 341, 487, 533 iStockphoto/ Th inkstock; p. 155 Opachevsky Irina/Shutterstock; p. 156 Chad Littlejohn/Shutterstock; p. 159 sahua d/Shutterstock; p. 206 Photos. com/Th inkstock; pp. 223, 419, 428 Mike van der Wold; p. 235 Picsfi ve/Shutterstock; p. 250 Vladislav Gurfi nkel/Shutterstock; p. 259 Mike Tan C.T./Shutterstock; p. 261 Suzan/Shutterstock; p. 261 R-studio/Shutterstock; pt. 264 Galyna Andrushko/Shutterstock; p. 299 Gustavo Miguel Fernandes/Shutterstock; p. 351 Stephanie Fray/Shutterstock; p. 375 Kristina Postnikova/Shutterstock; p. 397 Connors Bros./Shutterstock; p. 441 Philippe Wojazer/AP/Press Association Images f a r D © Cambridge University Press 2012 Acknowledgements vii IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd SSeecc33::vviiii 33//99//22001122 77::0000::1199 PPMM t f a r D © Cambridge University Press 2012 IIGGCCSSEE__MMaatthhss__PPrriilleemmss..iinndddd SSeecc33::vviiiiii 33//99//22001122 77::0000::1199 PPMM 11 Reviewing number concepts Key words • Natural number • Integer • Prime number t • Symbol • Multiple f • Factor • Composite numbers a • Prime factor • Square root • Cube • Directed numbers r • BODMAS D In this chapter you will learn how to: • identify and classify different types of numbers • fi nd common factors and common multiples of numbers • write numbers as products of their prime factors • calculate squares, square roots, cubes and cube roots of numbers • work with integers used in real-life situations • revise the basic rules for operating with numbers Th is statue is a replica of a 22 000-year-old bone found in the Congo. Th e real bone is only 10 cm long and it • is carved with groups of notches that represent numbers. One column lists the prime numbers from 10 to 20. perform basic calculations It is one of the earliest examples of a number system using tallies. using mental methods and with a calculator. Our modern number system is called the Hindu-Arabic system because it was developed by Hindus and spread by Arab traders who brought it with them when they moved to diff erent places in the world. Th e Hindu-Arabic system is decimal. Th is means it uses place value based on powers of ten. Any number at all, including decimals and fractions, can be written using place value and the digits from 0 to 9. © Cambridge University Press 2012 Unit 1: Number 1 IIGGCCSSEE__MMaatthhss__CChh0011__pp11--2211..iinndddd 11 33//88//22001122 55::0066::3355 PPMM