ICE-EM. Book 2 MATHEMATICS. Year7. Australian Curriculum Edition

Transcription

1 ICE-EM MATHEMATICS Australian Curriculum Edition 7 Year7 Book 2 Peter Brown Michael Evans Garth Gaudry David Hunt Janine McIntosh Bill Pender Jacqui Ramagge

2 C A M B R I D G E U N I V E R S I T Y P R E S S Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Information on this title: / The University of Melbourne on behalf of the Australian Mathematical Sciences Institute (AMSI) 2011 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 2011 Reprinted 2013 Edited by Joy Window Illustrated by Rob Mancini Typeset by Aptara Corp. Printed in Singapore by C.O.S. Printers Pte Ltd National Library of Australia Cataloguing in Publication data ICE-EM Mathematics. Year 7 book 2 / Peter Brown [et al.] Australian curriculum ed (pbk.) For secondary school age. Mathematics Study and teaching (Secondary) Australia. Mathematics Australia Textbooks. Brown, Peter Geoff. Australian Mathematical Sciences Institute ISBN Paperback ISBN Digital Additional resources for this publication at Reproduction and Communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this publication, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact: Copyright Agency Limited Level 15, 233 Castlereagh Street Sydney NSW 2000 Telephone: (02) Facsimile: (02) info@copyright.com.au Reproduction and Communication for other purposes Except as permitted under the Act (for example a fair dealing for the purposes of study, research, criticism or review) no part of this publication may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address above. 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 are correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

3 Contents Preface Acknowledgements Author biographies vii ix x Chapter 11 Integers 1 11A Negative integers 3 11B Addition and subtraction of a positive integer 7 11C Addition and subtraction of a negative integer 10 11D Multiplication involving negative integers 16 11E Division involving negative integers 19 11F Indices and order of operations 22 Review exercise 26 Challenge exercise 28 Chapter 12 Algebra and the Cartesian plane 30 12A Substitution with integers 32 12B The Cartesian plane 36 12C Completing tables and plotting points 42 12D Finding rules 46 Review exercise 52 Challenge exercise 54 Chapter 13 Triangles and constructions 56 13A Review of geometry 57 13B Angles in triangles 61 13C Circles and compasses 69 13D Isosceles and equilateral triangles 73 13E Constructions with compasses and straight edge 79 13F Quadrilaterals 83 13G Further constructions 87 Review exercise 88 Challenge exercise 91 iii

4 Chapter 14 Negative fractions and decimals 93 14A Addition and subtraction of negative fractions 95 14B Multiplication and division of negative fractions C Negative decimals D Substitution involving negative fractions and decimals 109 Review exercise 112 Challenge exercise 114 Chapter 15 Percentages and ratios A Percentages, fractions and decimals B One quantity as a percentage of another C Percentage of a quantity D Ratios E Solving problems with ratios F Best buys 134 Review exercise 136 Challenge exercise 138 Chapter 16 Solving equations A An introduction to equations B Equivalent equations C Solving equations involving more than one step D Equations with negative solutions E Expanding brackets and solving equations F Collecting like terms and solving equations G Equations with pronumerals on both sides H Solving problems using equations 163 Review exercise 167 Challenge exercise 170 iv

5 Chapter 17 Probability A An introduction to probability B Experiments and counting 175 Review exercise 184 Challenge exercise 186 Chapter 18 Transformations and symmetry A Translations B Rotations C Reflections D Combinations of transformations E Transformations in the Cartesian plane F Symmetry G Regular polygons 217 Review exercise 221 Challenge exercise 225 Chapter 19 Graphs and tables A Reading tables B The pictogram C Column graphs D Divided bar charts and pie charts E Line graphs F Applications of the line graph 257 Review exercise 265 Chapter 20 Statistics A Data and dot plots B The mode C Stem-and-leaf plots D Median, mean and range 282 Review exercise 289 v

6 Chapter 21 Polyhedra and three-dimensional drawing A Polyhedra B Drawing a solid C Perspective drawing 306 Review exercise 310 Challenge exercise 311 Chapter 22 Review and problem-solving A Review B Tessellations C Sets and Venn diagrams 331 Answers to exercises 354 vi

7 Preface ICE-EM Mathematics is a series of textbooks for students in years 5 to 10 throughout Australia who study the Australian Mathematics Curriculum. Background The International Centre of Excellence for Education in Mathematics (ICE-EM) was established in 2004 with the assistance of the Australian Government and is managed by the Australian Mathematical Sciences Institute (AMSI). The Centre originally published the series as part of a program to improve mathematics teaching and learning in Australia. AMSI is now collaborating with Cambridge University Press to publish Australian Curriculum editions of the series. ICE-EM developed the program and textbooks in recognition of the importance of mathematics in modern society and the need to enhance the mathematical capabilities of Australian students. Students who use the series will have a strong foundation for work or further study. Features ICE-EM Mathematics provides a progressive development from upper primary to middle secondary school. The year 10 textbooks incorporate all material for the 10A course, and selected topics in earlier books carefully prepare students for this. ICE-EM Mathematics is an excellent preparation for all of the Australian Curriculum s year 11 and 12 mathematics courses. The writers of the series are some of Australia s most outstanding mathematics teachers and subject experts. The textbooks are clearly and carefully written, and contain background information, examples and worked problems. Each chapter addresses a specific Australian Curriculum content strand and set of sub-strands. The exercises within chapters take an integrated approach to the concept of proficiency strands, rather than separating them out. Students are encouraged to develop and apply understanding, fluency, problem-solving and reasoning skills in every exercise. The series places a strong emphasis on understanding basic ideas, along with mastering essential technical skills. Mental arithmetic and other mental processes are major focuses, as is the development of spatial intuition, logical reasoning and understanding of the concepts. Problem-solving lies at the heart of mathematics, so ICE-EM Mathematics gives students a variety of different types of problems to work on, which help them develop their reasoning skills. Challenge exercises at the end of each chapter contain problems and investigations of varying difficulty that should catch the imagination and interest of students. The final chapter in each 7 10 textbook contains additional problems that cover new concepts for students who wish to explore the subject even further. vii

8 The problems and examples in the ICE-EM Mathematics series are written in a way that deliberately does not require the use of a calculator, except in appropriate contexts, until year 9. During primary and early secondary years, students need to become confident with mental and written calculations, using a variety of techniques. These skills are essential to students mathematical development, and lead to a feeling of confidence and mathematical self-reliance. Furthermore, as different states have varying requirements and expectations of calculator use, the series is designed to be as calculator neutral as possible. Additional resources Cambridge HOTmaths provides an integrated program for users of the ICE-EM Mathematics series, combining the best of textbook and interactive online resources. The presence of a HOTmaths icon (shown at the right) in the header of a chapter topic shows that resources are available for that topic. Materials are accessible from a dropdown menu in HOTmaths, organised by textbook chapter and topic/lesson structure. For more information, see The ICE-EM Mathematics website at provides further support materials for teachers and students, as well as links to supplementary and enrichment materials. There are also teacher resources for the Foundation to Year 10 Australian Mathematics Curriculum available at You can read more about AMSI at and also see the mathematics involved in a variety of careers at viii

9 Acknowledgements We are grateful to Professor Peter Taylor, Director of the Australian Mathematics Trust, for his support and guidance as chairman of the Australian Mathematical Sciences Institute Education Advisory Committee. We gratefully acknowledge the major contribution made by those schools that participated in the Pilot Program during the development of the ICE-EM Mathematics program. We also gratefully acknowledge the assistance of: Josian Astruc Sue Avery Robyn Bailey Brian Dorofaeff Andy Edwards Claire Ho Nikolas Sakellaropoulos James Wan The author and publisher wish to thank the following sources for permission to reproduce material: Images: Cambridge University Press/ Nicole Tuck, p. 181, 185; Frans Hals, Portrait du philosophe René Descartes ( ), p. 36; 2011 Used under license from Shutterstock. com/ VanHart, p. 31/ AlexSmith, p. 140, 141, 157 (pencil case)/ violetkaipa, p. 140, 141 (pencils)/ gjfoto, p. 172/ Dimitar Bosakov, p. 173/ cristi180884, p. 175/ luchschen, p. 274/ takito, p Every effort has been made to trace and acknowledge copyright. The publisher apologises for any accidental infringement and welcomes information that would redress this situation. ix

10 Author biographies Peter Brown Peter Brown studied Pure Mathematics and Ancient Greek at Newcastle University, and completed postgraduate degrees in each subject at the University of Sydney. He worked for nine years as a mathematics teacher in NSW State schools. Since 1990, he has taught Pure Mathematics at the School of Mathematics and Statistics at the University of New South Wales (UNSW). He was appointed Director of First Year Studies in He specialises in Number Theory and History of Mathematics and has published in both areas. Peter regularly speaks at teacher inservices, Talented Student days and Mathematics Olympiad Camps. In 2008 he received a UNSW Vice Chancellor s Teaching Award for educational leadership. Michael Evans Michael Evans has a PhD in Mathematics from Monash University and a Diploma of Education from La Trobe University. He is currently employed at the Australian Mathematics Sciences Institute (AMSI). Before this, he was Head of Mathematics at Scotch College, Melbourne. He has also taught in public schools and he has been involved with curriculum development and assessment in Victoria for many years. In 1999, Michael was awarded an honorary Doctor of Laws by Monash University for his contribution to mathematics education, and in 2001 he received the Bernhard Neumann Award for contributions to mathematics enrichment in Australia. Garth Gaudry Garth Gaudry was Head of Mathematics at Flinders University before moving to UNSW, where he became Head of School. He was the inaugural Director of AMSI before becoming Director of AMSI s International Centre of Excellence for Education in Mathematics. Previous positions include membership of the South Australian Mathematics Subject Committee and the Eltis Committee appointed by the NSW Government to inquire into Outcomes and Profiles. He is a life member of the Australian Mathematical Society and Emeritus Professor of Mathematics, UNSW. David Hunt David Hunt graduated from the University of Sydney in 1967 with an Honours degree in Mathematics and Physics, then obtained a master s degree and a doctorate from the University of Warwick. He was appointed to a lectureship in Pure Mathematics at UNSW in early 1971, where he is currently an Associate Professor. David has taught courses in Pure Mathematics from first year to master s level and was Director of First Year Studies in Mathematics for five years. Many of David s activities outside UNSW have centred on the Australian Mathematics Trust. He is currently Deputy Chairman of the Australian Mathematics Olympiad Committee. x

11 Janine McIntosh Janine McIntosh works at the Australian Mathematical Sciences Institute where her role is to write mathematics materials and to work with teachers to develop their mathematics programs. Janine is an experienced primary teacher, curriculum writer and teacher educator. Bill Pender Bill Pender has a PhD in Pure Mathematics from Sydney University and a BA (Hons) in Early English from Macquarie University. After a year at Bonn University, he taught at Sydney Grammar School from 1975 to 2008, where he was Subject Master for many years. He has been involved in the development of NSW Mathematics syllabuses since the early 1990s, and was a foundation member of the Education Advisory Committee of AMSI. He has also lectured and tutored at Sydney University and at UNSW, and given various inservice courses. Bill is the lead author of the NSW calculus series Cambridge Mathematics. Jacqui Ramagge Jacqui Ramagge is currently Head of the School of Mathematics and Applied Statistics at the University of Wollongong (UOW) and is a member of the Engineering, Mathematics and Informatics panel of the Australian Research Council College of Experts. After graduating in 1993 with a PhD in Mathematics from the University of Warwick (UK), she worked at the University of Newcastle (Australia) until 2007, when she moved to UOW. She teaches mathematics at all university levels, is part of the CSIRO Mathematicians in Schools program, and has won a teaching award. She contributed to the Vermont Mathematics Initiative (USA) and is a founding member of the Australian Mathematics Trust Primary Problems Committee. xi

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