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Alan McSevenyRob ConwaySteve WilkesMichael Smith4Mathematicsfor the Middle YearsInternational inter_maths4_title.indd 1 17/4/09 2:27:29 PMPearson Australia(a division of Pearson Australia Group Pty Ltd)20 Thackray Road, Port Melbourne, Victoria 3207PO Box 460, Port Melbourne, Victoria 3207 Australiawww.pearson.com.auOffices in Sydney, Brisbane and Perth, and associated companies throughout the world.Copyright Pearson Australia 2009First published 2009All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.This book is not to be treated as a blackline master; that is, any photocopying beyond fair dealing requires prior written permission.Editor: Fiona BrodribbDesigner: Pier Vido and Anita AdamsCopyright and Pictures Editor: Jacqui LiggetIllustrator/s: Michael Barter, Bruce Rankin and Boris SilvestriTypeset by Sunset Digital Pty Ltd, BrisbanePrinted in ChinaNational Library of AustraliaCataloguing-in-Publication dataInternational mathematics for the middle years 41st ed.Includes index.For secondary school students.ISBN 9780733985195 (pbk: CD-ROM)1. Mathematics Textbooks. I. McSeveny, A. (Alan).510Pearson Australia Group Pty Ltd ABN 40 004 245 943 Acknowledgements Pearson Australia would like to thank Luke Borda for his assistance in reviewing and writing some of the material in International Mathematics for the Middle Years Book 4 . Luke is the Mathematics Coordinator for the Middle and Senior Schools at Mercedes College in Adelaide, Australia, where he has been teaching MYP mathematics for the last 5 years. Luke has been involved in submitting year 5 MYP moderation and has focussed on the assessment criteria and their application and interpretation.Every effort has been made to trace and acknowledge copyright. However, should any infringement have occurred, the publishers tender their apologies and invite copyright owners to contact them. IM4_Ch00_Prelims.fmPage iiMonday, March 16, 20093:44 PM iii Contents Features of International Mathematics for the Middle Years viiiUsing this Book for Teaching MYPfor the IB xiiMetric Equivalents xiiiThe Language of Mathematics xiv ID Card 1 (Metric Units) xivID Card 2 (Symbols) xivID Card 3 (Language) xvID Card 4 (Language) xviID Card 5 (Language) xviiID Card 6 (Language) xviiiID Card 7 (Language) xix Algebra Card xxBasic Skills and Number Review 1 1:01 Basic number skills 21:02 Algebraic expressions 71:03 Probability 71:04 Geometry 81:05 Indices 91:06 Measurement 101:07 Equations and inequations 111:08 Coordinate geometry 121:09 Statistics 131:10 Formulae and problem-solving 141:11 Graphs of physical phenomena 15 Working mathematically 16 Proportion 17 2:01 Review Proportional change 182:02 A new approach 20 A proportional flip 22 2:03 Inverse proportion (inverse variation) 24 Mathematical terms Diagnostic test Revision assignment 26 Consumer Arithmetic 29 3:01 Working for others 303:02 Extra payments 34 Jobs in the papers 37 3:03 Wage deductions 383:04 Taxation 42 What is brought to the table, cut, but never eaten? 44 3:05 Budgeting 453:06 Best buy, shopping lists and change 473:07 Sales tax 51 Shopper dockets 53 3:08 Ways of paying and discounts 54 The puzzle of the missing dollar 58 3:09 Working for a profit 59 Lets plan a disco 62Mathematical terms Diagnostic test Revision assignment Working mathematically 62 Algebraic Expressions 68 4:01 Generalised arithmetic 69 Lets play with blocks 72 4:02 Substitution 73 The history of algebra 74 4:03 Simplifying algebraic expressions 744:04 Algebraic fractions 76 A Addition and subtraction 76 B Multiplication and division 78 Try this maths-word puzzle 79 4:05 Simplifying expressions with grouping symbols 80 What is taken off last before you get into bed? 82 4:06 Binomial products 834:07 Special products 85 A Perfect squares 85 The square of a binomial 85B Difference of two squares 874:08 Miscellaneous examples 88 Patterns in products 89Using special products in arithmetic 90Mathematical terms Diagnostic test Revision assignment Working mathematically 91 Indices and Surds 96 5:01 Indices and the index laws 975:02 Negative indices 1015:03 Fractional indices 103 Why is a room full of married people always empty? 106Reasoning with fractional indices 106 5:04 Scientific (or standard) notation 108 Multiplying and dividing by powers of 10 108 5:05 Scientific notation and the calculator 112 Using scientific notation 114 5:06 The real number system 115 Proof that is irrational 117f-stops and118 5:07 Surds 1195:08 Addition and subtraction of surds 1215:09 Multiplication and division of surds 123 Iteration to find square roots 125Chapter 1Chapter 2Chapter 3Chapter 4Chapter 522 IM4_Ch00_Prelims.fmPage iiiMonday, March 16, 20093:44 PM iv INTERNATIONAL MATHEMATICS 4 5:10 Binomial products 1265:11 Rationalising the denominator 128 What do Eskimos sing at birthday parties? 129Rationalising binomial denominators 130Mathematical terms Diagnostic test Revision assignment Working mathematically 131 Equations, Inequations and Formulae 136 6:01 Equivalent equations 1376:02 Equations with grouping symbols 140 Solving equations using a spreadsheet 142 6:03 Equations with fractions (1) 1436:04 Equations with fractions (2) 145 Equations with pronumerals in the denominator 147 6:05 Solving problems using equations 1486:06 Inequations 153 Solutions to inequations 153Read carefully (and think)! 157 6:07 Formulae: Evaluating the subject 1586:08 Formulae: Equations arising from substitution 1606:09 Solving literal equations (1) 1626:10 Solving literal equations (2) 1646:11 Solving problems with formulae 167 Mathematical terms Diagnostic test Revision assignment Working mathematically 170 Factorising Algebraic Expressions 175 7:01 Factorising using common factors 1767:02 Factorising by grouping in pairs 1787:03 Factorising using the difference of two squares 180 The difference of two cubes (Extension) 181 7:04 Factorising quadratic trinomials 182 How much logic do you have? 183 7:05 Factorising further quadratic trinomials 184 Another factorising method for harder trinomials 187 7:06 Factorising: Miscellaneous types 188 What did the caterpillar say when it saw the butterfly? 189 7:07 Simplifying algebraic fractions: Multiplication and division 1907:08 Addition and subtraction of algebraic fractions 193 Mathematical terms Diagnostic test Revision assignment Working mathematically 196 Coordinate Geometry 200 8:01 The distance between two points 2018:02 The midpoint of an interval 2068:03 The gradient of a line 2108:04 Graphing straight lines 2158:05 The gradientintercept form of a straight line: y = mx + c 220 What does y = mx + c tell us? 220 8:06 The equation of a straight line, given pointand gradient 2258:07 The equation of a straight line, giventwo points 2278:08 Parallel and perpendicular lines 2308:09 Graphing inequalities on the number plane 234 Why did the banana go out with a fig? 238Mathematical terms Diagnostic test Revision assignment Working mathematically 239 Simultaneous Equations 244 Solving problems by guess and check 245 9:01 The graphical method of solution 246 Solving simultaneous equations using agraphics calculator 250What did the book say to the librarian? 250 9:02 The algebraic method of solution 251 A Substitution method 251 B Elimination method 2539:03 Using simultaneous equations to solveproblems 256 Breakfast time 259Mathematical terms Diagnostic test Revision assignment Working mathematically 260 Graphs of Physical Phenomena 263 10:01 Distance/time graphs 264 A Review of linear graphs 264 Graphing coins 267B Non-linear graphs 268 Rolling down an inclined plane 274 10:02 Relating graphs to physical phenomena 275 Spreadsheet graphs 284Make words with your calculator 285Curves and stopping distances 286Mathematical terms Diagnostic test Revision assignment Working mathematically 287 Deductive Geometry 293 11:01 Deductive reasoning in numerical exercises 294 A Exercises using parallel lines 294 B Exercises using triangles 296 C Exercises using quadrilaterals 29811:02 Review of polygons 300 Regular polygons and tessellations 302Spreadsheet 304The game of Hex 305 11:03 Deductive reasoning in non-numerical exercises 30611:04 Congruent triangles 309Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11 IM4_Ch00_Prelims.fmPage ivMonday, March 16, 20093:44 PM v 11:05 Proving two triangles congruent 31311:06 Using congruent triangles to find unknownsides and angles 31711:07 Deductive geometry and triangles 31911:08 Deductive geometry and quadrilaterals 323 Theorems and their converses 328What do you call a man with a shovel? 328 11:09 Pythagoras theorem and its converse 329 Proving Pythagoras theorem 329Mathematical terms Diagnostic test Revision assignment Working mathematically 332 Measurement 337 12:01 Review Area and perimeter 338 Staggered starts 339Skirting board and perimeter 340Conversions 346Covering floors 350 12:02 Review of surface area of prisms 35112:03 Volume of prisms and cylinders 357 Tank sizes 359 12:04 Volume of prisms, cylinders and composite solids 360 Perimeter, area and volume 364 12:05 Practical applications of measurement 366 Wallpapering rooms 368Mathematical terms Diagnostic test Revision assignment Working mathematically 371 Trigonometry 376 13:01 Right-angled triangles 37713:02 Right-angled triangles: the ratio of sides 37913:03 The trigonometric ratios 38113:04 Trig. ratios and the calculator 387 The exact values for the trig. ratio of30, 60 and 45 390 13:05 Finding an unknown side 39113:06 Finding an unknown angle 39713:07 Miscellaneous exercises 40013:08 Problems involving two right triangles 405 What small rivers flow into the Nile? 408Mathematical