Dynamic Learning is an online subscription solution that supports teachers and students with quality content and unique tools. Dynamic Learning incorporates elements that all work together to give the ultimate classroom and homework resources. Teaching and Learning Resources titles include resources, lesson planning tools, self-marking tests and assessments. Teachers can: ● Use the Lesson Builder to plan and deliver lessons ● Share lessons and resources with students and colleagues ● Track student’s progress Teachers can also combine their own trusted resources alongside those from Edexcel GCSE (9–1) Mathematics Success in a Year which has a whole host of informative, exciting and interactive resources including: ● Exam practice papers ● Diagnostic tests that highlight misconceptions and help to reinforce learning ● Follow-up Assessments (knowledge) and Follow-up Assessments (problem-solving) ● Learning Exercises ● Problem-solving Exercises and Problem-solving Strategies ● Interactive Learning Videos ● Interactive Worked Examples Edexcel GCSE (9–1) Mathematics Success in a Year is also available as a Whiteboard eTextbook which is an online interactive version of the printed textbook that enables teachers to: ● Display interactive pages to their class ● Add notes and highlight areas ● Add double-page spreads into lesson plans Additionally the Student eTextbook of Edexcel GCSE (9–1) Mathematics Success in a Year is a downloadable version of the printed textbook that teachers can assign to students so they can: ● Download and view on any device or browser ● Add, edit and synchronise notes across two devices ● Access their personal copy on the move To fi nd out more and sign up for free trials visit: www.hoddereducation.co.uk/dynamiclearning MASTERING MATHEMATICS EDEXCEL GCSE MATHEMATICS Success in a Year Heather Davis Series Editor Elaine Lambert In order to ensure that this resource offers high-quality support for the associated Pearson qualification, it has been through a review process by the awarding body. 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Although every effort has been made to ensure that website addresses are correct at time of going to press, Hodder Education cannot be held responsible for the content of any website mentioned in this book. It is sometimes possible to find a relocated web page by typing in the address of the home page for a website in the URL window of your browser. Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders: please contact Bookpoint Ltd, 130 Milton Park, Abingdon, Oxon OX14 4SB. Telephone: +44 (0)1235 827720. Fax: +44 (0)1235 400454. Lines are open 9.00a.m.–5.00p.m., Monday to Saturday, with a 24-hour message answering service. Visit our website at www.hoddereducation.co.uk © Gareth Cole, Cornwall Council, Heather Davis, Sophie Goldie, Hodder & Stoughton, Elaine Lambert, Linda Liggett, Robin Liggett, Andrew Manning, Richard Perring, Keith Pledger, Roger Porkess, South Dartmoor Community College, Rob Summerson, Kevin Tanner, Colin White 2017 First published in 2017 by Hodder Education, An Hachette UK Company Carmelite House 50 Victoria Embankment London EC4Y 0DZ Impression number 10 9 8 7 6 5 4 3 2 1 Year 2020 2019 2018 2017 All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Cover photo © Four sided triangle/Alamy Stock Photo Typeset in Humanist 521 BT Light 11/12 by Integra Software Services Pvt. Ltd., Pondicherry, India Printed in Italy A catalogue record for this title is available from the British Library ISBN 978 1471 886775 Contents Each chapter number corresponds to a topic. Most topics appear in both the Essential Topics and Next Steps sections, but some only appear in one section. Therefore, Essential Topics has no Chapter 12 and no Chapter 19 as these are only included in Next Steps. Likewise there are no Chapters 1, 4, 5,16, 20 or 21 in Next Steps as these topics are only in Essential Topics. How to get the most from this book viii Essential Topics NUMBER Chapter 1 Calculating 1 1.1 Order of operations 4 1.2 Multiplying decimals 8 1.3 Dividing decimals 12 Chapter 2 Using our number system 17 2.1 Using the number system effectively 20 2.2 Understanding standard form 24 Chapter 3 Accuracy 28 3.1 Signifi cance 30 3.2 Approximating 33 Chapter 4 Fractions 39 4.1 Working with mixed numbers 43 4.2 Dividing fractions 48 Chapter 5 Percentages 53 5.1 Applying percentage increases and decreases to amounts 56 5.2 Finding the percentage change from one amount to another 61 5.3 Reverse percentages 66 5.4 Repeated percentage increase/decrease 70 Chapter 6 Ratio and proportion 75 6.1 Sharing in a given ratio 75 6.2 Working with proportional quantities 79 iii s Chapter 7 Number properties 85 t n 7.1 Index notation 87 e t n 7.2 Prime factorisation 90 o C AlgebrA Chapter 8 Starting algebra 96 8.1 Working with more complex equations 101 8.2 Solving equations with brackets 107 Chapter 9 Sequences 112 9.1 Linear sequences 114 9.2 Special sequences 120 Chapter 10 Functions and graphs 126 10.1 The equation of a straight line 129 10.2 Plotting quadratic and cubic graphs 137 Chapter 11 Algebraic methods 146 11.1 Linear inequalities 146 11.2 Solving pairs of equations by substitution 152 11.3 Solving simultaneous equations by elimination 157 geometry ANd meASureS Chapter 13 units and scales 163 13.1 Scale drawing 169 13.2 Compound units 174 Chapter 14 Properties of shapes 178 14.1 Angles in parallel lines 182 14.2 Angles in a polygon 189 Chapter 15 measuring shapes 194 15.1 Area of circles 198 15.2 Pythagoras’ theorem 204 Chapter 16 Construction 211 16.1 Constructions with a pair of compasses 212 16.2 Loci 222 Chapter 17 transformations 230 17.1 Similarity 237 Chapter 18 three-dimensional shapes 245 18.1 Prisms 250 18.2 Constructing plans and elevations 258 iv StAtiStiCS ANd ProbAbility C o n Chapter 20 Statistical measures 264 t e n 20.1 Using grouped frequency tables 267 t s Chapter 21 Statistical diagrams 275 21.1 Pie charts 278 21.2 Displaying grouped data 284 21.3 Scatter diagrams 292 21.4 Use and limits of lines of best fit 298 Chapter 22 Probability 304 22.1 Combined events 307 miNi teSt PAPerS essential topics test Paper 1 315 essential topics test Paper 2 318 essential topics test Paper 3 321 Problem Solving Chapter 1 getting started 324 Chapter 2 making a plan 331 Chapter 3 Carrying out the plan 341 Chapter 4 reviewing the solution 346 Chapter 5 bringing it all together 351 Next Steps Number Chapter 2 using our number system 358 2.3 Calculating with standard form 358 Chapter 3 Accuracy 363 3.3 Limits of accuracy 363 Chapter 6 ratio and proportion 367 6.3 The constant of proportionality 367 6.4 Working with inversely proportional quantities 373 Chapter 7 Number properties 378 7.3 Rules of indices 378 v s AlgebrA t n e Chapter 8 Starting algebra 381 t n o 8.3 Simplifying harder expressions 381 C 8.4 Using complex formulae 385 8.5 Identities 389 Chapter 9 Sequences 393 9.3 Quadratic sequences 393 9.4 Geometric progressions 399 Chapter 10 Functions and graphs 403 10.3 Finding equations of straight lines 403 10.4 Quadratic functions 408 10.5 Polynomial and reciprocal functions 414 Chapter 11 Algebraic methods 421 11.4 Using graphs to solve simultaneous equations 421 Chapter 12 Working with quadratics 427 12.1 Factorising quadratics 427 12.2 Solving equations by factorising 431 geometry ANd meASureS Chapter 13 units and scales 435 13.3 Working with compound units 435 Chapter 14 Properties of shapes 441 14.3 Congruent triangles and proof 441 14.4 Proof using similar and congruent triangles 448 Chapter 15 measuring shapes 453 15.3 Arcs and sectors 453 Chapter 17 transformations 460 17.2 Trigonometry 460 17.3 Trigonometry for special angles 467 17.4 Finding centres of rotation 470 Chapter 18 three-dimensional shapes 476 18.3 Surface area and volume of 3-D shapes 476 Chapter 19 Vectors 480 19.1 Vectors 480 vi StAtiStiCS ANd ProbAbility C o n Chapter 22 Probability 486 t e n 22.2 Estimating probability 486 t s 22.3 The multiplication rule 492 22.4 The addition rule 500 miNi teSt PAPerS Next Steps test Paper 1 506 Next Steps test Paper 2 509 Next Steps test Paper 3 512 index 515 Answers to all questions in the book are provided online at www.hoddereducation.co.uk/ EdexcelGCSESuccessInAYear vii How to get the most from this book This book covers the content to gain a grade 5 in the GCSE (9–1) specifi cation for Edexcel. The material is split into three parts: • Essential Topics • Problem Solving • Next Steps Essential Topics Start with thAed Edisnsge anntdia sl uTbotpraiccsti.n Tg hfreascet ioanres topics that you have probably met beCfore, and you will H have already mastered some of them. They all need to be mastered. A Work throuJagrehd htahs 1i1s72 pof ahris tc askee qlefut.entially, chapter by chapter. PTER • Use theHH Seo wgki vmeislu lc2sh o isfc tlhehfet encaockwek ?to –hi s wsistaer.rming up section to uncover any misconceptio 4 FRns in your prior A knowledge thEatS mSaEyN stTopIA yLo uT mOaPsIteCrSin g– t hNeU chMapBteErR. If you need a quick remCinder, look at the T APTER Jm0ues3tth iond scSJa.aro eIlsdfu ehytaio osAs n1ue72 c c–nt c12eio lueenftd.r, aeloxcctaryate hde blpe fworiteh tahney s okifl ltsh cehseec wk,a wrmhe-urep ytoopui cws,i lwl fi onrdk tthherIONSo buagshic tfhaect sr ealenvda nt H C online re7so – u1 r=c e7s .– 6 = 1 12 2 12 12 12 1 of the cake is left. 12 JUST IN CAS×E6 Rounding1 = 6 1 and 6 are equivalent fractions. 2 12 2 12 Rounding means writing a number less accurately. It can help to make numbers more understandable and mana geable. ×6 Rounding to the nearest 10 means you write down the nearest multiple of 10 to the number. Numbers tshaamt ea r1te7h2 itnhg e–; tsha21emr ee= ids i s1nt7oa2n mc e–a tfhr1o6e2mm at=twic oa1l 1 m2reua lstiopnle fso or fi t1.0, Oeynnodciuen g ct ahinne 5 sd,u erbnotouranmcdit n utapht.eo T rthso iasp r isne ustomh ewb esear amsl.l ed,o the Here is a number line. Circle the whole numbers that round to 160 to the nearest 10. 12 is the lowest common denominator 150151152153154155156157158159160161162163164165166o1f6 1721 6a8nd16 29.170 Notice that 165 rounds Solution to 170, not 160. 150151152153154155156157158159160161162163164165166167168169170 Rounding to one decimal place You cSanK roIunLd LtoS an yC nuHmbEerC buKt u sually you will be asked to round to a power of 10, such as 1000, 100, 10 or 1 whole, or to a number of decimal places. Round each of these numbers on this number line to 1 decimal place. a 2.127 b 2(cid:31).473 6Warming up c 3.0491758 Answers to all Comaplete the skills cbhecks below to make sure ycou are ready for this chapter. Further revision of this assumed knowledge is available in the online Dynamic Learning Resources. questions in the book are provided 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 (cid:31) Understanding fractions online at www. Work out each of these. hoddereducation. a 1 of £24 b 3 of 84 people co.uk/EdexcelGCSE 3 4 28 c 5 of 16 ounces d 1 of £16 SuccessInAYear 8 4 viii 4411