mathscape 10 ext prelims

Mathscape 10 ext. - Prelims Page i Tuesday, October 4, 2005 3:52 PM

First published 2005 by

MACMILLAN

EDUCATION

AUSTRALIA

PTY

LTD

627 Chapel Street, South Yarra 3141

Visit our website at www.macmillan.com.au

Associated companies and representativesthroughout the world.

Copyright © Clive Meyers, Graham Barnsley, Lloyd Dawe, Lindsay Grimison 2005

All rights reserved.Except under the conditions described in the Copyright Act 1968 of Australia (the Act) 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 written permission of the copyright owner.

Educational institutions copying any part of this book for educational purposes under the Act must be covered by a Copyright Agency Limited (CAL) licence for educational institutionsand must have given a remuneration notice to CAL. Licence restrictions must be adhered to. For details of the CAL licence contact: Copyright Agency Limited, Level 19, 157 Liverpool Street, Sydney, NSW 2000. Telephone: (02) 9394 7600. Facsimile: (02) 9394 7601. Email: [email protected]

National Library of Australiacataloguing in publication data

Meyers, Clive.Mathscape 10 extension.

For secondary school students.ISBN 0 7329 8087 9.

1. Mathematics – Textbooks. I. Grimison, Lindsay. II.Barnsley, Graham J. III. Dawe, Lloyd. IV. Title.

510

Publisher: Ben DaweProject editor: Colin McNeilEditor: Colin McNeilIllustrator: Stephen FrancisCover and text designer: Dimitrios FrangoulisTypeset in 11/13 pt Times by Sun Photoset Pty LtdCover image: Photolibrary.com/Science Photo Library/John Mead

Printed in Malaysia

Internet addresses

At the time of printing, the Internet addresses appearing in this book were correct. Owing to the dynamic nature of the Internet, however, we cannot guarantee that all these addresses will remain correct.

Publisher’s acknowledgments

The authors and publishers would like to gratefully credit or acknowledge the following for permission to reproduce copyright material:

AAP, p. 277; APL/Corbis/Roger Ressmeyer, p. 242; British Film Institute, p. 214; Corbis, pp. 1–27; Digital Stock, pp. 32–68, 195–216, 219–79, 423–56; Digital Vision, p. 110; Fairfax Photos/Angela Wylie, p. 157, /Peter Rae, p. 326; Getty Images/Jessica Gow/AFP, p. 65, /Tony Hallas, p. 416, /STR/AFP, p. 490, /William West/AFP, p. 369; Istockphoto.com/Galina Barskaya, pp. 531–77, /Henry Kippert, pp. 498–528; Photodisc, pp. 72–113, 118–59, 164–92, 285–331, 339–71, 377–419, 460–94; Photolibrary/Kindra Clineff, p. 190; Rob Cruse Photography, pp. 17, 25; The Art Archive, p.526. The following ‘Try This’ activities are text extracts from

New Course Mathematics 10 Advanced

by Paul Bigelow and Graeme Stone, Macmillan Education Australia, 1998, pp. 60, 93, 97, 132, 138, 148, 157, 207, 213, 242, 264, 390, 415, 437, 440, 449, 453, 468, 485, 490, 504, 543, 553, 561; Text extracts from

New Course Mathematics 10 Advanced

by Paul Bigelow and Graeme Stone, Macmillan Education Australia, 1998, pp. 280, 444, 453; The following ‘Language Links’ are text extracts from Macquarie Learner’s Dictionary, Macquarie Library, 1999, pp. 28, 68, 113, 160, 192, 279, 332, 372, 420, 457, 495.

While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable arrangement with the rightful owner in each case.

Mathscape 10 ext. - Prelims Page ii Tuesday, October 4, 2005 3:52 PM

Contents

Preface vi

How to use this book vii

Chapter 1 Consumer arithmetic 11.1 Simple interest 21.2 Compound interest 4

Try this: Inflation 91.3 Depreciation 9

Try this: Depreciating value 131.4 Buying major items 131.5 Credit cards 171.6 Loans 21

Try this: Housing loans 25Focus on working mathematically:

The cost of living in Australia 25Language link with Macquarie 28Chapter review 28

Chapter 2 Trigonometry 322.1 The trigonometric ratios 332.2 Degrees and minutes 352.3 Finding the length of a side 38

Try this: Swing 452.4 Finding the size of an angle 45

Try this: High flying 502.5 The tangent ratio 502.6 The complementary results 532.7 The exact values 55

Try this: Square Area 602.8 Bearings review 60Focus on working mathematically:

Orienteering 65Language link with Macquarie 68Chapter review 69

Chapter 3 Volume and surface area 723.1 Surface area of a prism 733.2 Surface area of a pyramid 763.3 Surface area of a cylinder 79

Try this: Elephants 823.4 Surface area of a cone and sphere 833.5 Volume of a prism 88

Try this: Can we use an iceberg? 93

3.6 Volume of a cylinder 94Try this: Magic nails? 97

3.7 Volume of a pyramid 97Try this: Popcorn 102

3.8 Volume of a cone 1023.9 Volume of a sphere 106

Try this: Cones, spheres and cylinders 109

Focus on working mathematically:The surface area of a soccer ball 110

Language link with Macquarie 113Chapter review 114

Chapter 4 Deductive geometry 1184.1 Simple numerical exercises 1194.2 Polygons 129

Try this: Ratio of exterior angles 1324.3 Harder numerical problems 1324.4 Deductive proofs involving angles 135

Try this: Angles in a rhombus 1384.5 Deductive proofs involving

triangles 1384.6 Congruent triangles review 142

Try this: Intersecting parallelograms 148

4.7 Deductive proofs involving quadrilaterals 148

4.8 Pythagoras’ Theorem 152Try this: An unusual proof of Pythagoras’ theorem 157

Focus on working mathematically:An exploration of an equilateral

triangle 157Language link with Macquarie 160Chapter review 160

Chapter 5 Factorisation and algebraic fractions 1645.1 Binomial products review 1655.2 The highest common factor 169

Try this: Market garden 1715.3 Difference of two squares 1715.4 Grouping in pairs 173

Try this: Squaring fives 1755.5 Factorising monic quadratic

trinomials 175

iii

Mathscape 10 ext. - Prelims Page iii Tuesday, October 4, 2005 3:52 PM

Mathscape 10iv

5.6 Factorising general quadratic trinomials 177

5.7 Miscellaneous factorisations 181Try this: Difference of two squares 182

5.8 Simplifying algebraic fractions 1825.9 Multiplying and dividing algebraic

fractions 1845.10 Adding and subtracting algebraic

fractions 187Focus on working mathematically:

Taxicab numbers and the sum of two cubes 190


Chapter 6 Quadratic equations 1956.1 Quadratic relationships 1966.2 Solving quadratic equations using

factors 199Try this: Forming a quadraticequation 201

6.3 Completing the square 2026.4 The quadratic formula 204

Try this: Chocolate time 2076.5 Miscellaneous equations 2076.6 Practical problems involving

quadratic equations 210Try this: Touching circles 213

Focus on working mathematically:The long distance flight of arrows

under gravity 214Language link with Macquarie 216Chapter review 217

Chapter 7 Graphs in thenumber plane 2197.1 Graphs of physical phenomena 2207.2 Travel graphs 2277.3 Straight line graphs 2317.4 The parabola y = ax2 239

Try this: The Parkes telescope 2427.5 The parabola y = ax2 + c 243

Try this: Flagpoles 2477.6 The parabola y = ax2 + bx + c 2477.7 The axis of symmetry of the

parabola 2507.8 The parabola y = (x − h)2 + k 2537.9 The cubic curve y = ax3 + d 257

7.10 The hyperbola 260Try this: Gas under pressure 264

7.11 The exponential curve 2647.12 The circle 2677.13 Miscellaneous graphs 270

Try this: Maximum area 277Focus on working mathematically:

The Tower of Terror 277Language link with Macquarie 279Chapter review 280

Chapter 8 Data analysis andevaluation 2858.1 Data analysis review 286

Try this: Misleading graphs 2988.2 The interquartile range 298

Try this: Cliometrics 3048.3 The standard deviation 3048.4 Applications of the standard

deviation 309Try this: Two standard deviations 314

8.5 The shape of a distribution 315Try this: Correlation 319

8.6 Comparing two data sets 319Focus on working mathematically:

The imprisonment of Indigenouspeople 326


Chapter 9 Probability 3389.1 Probability review 3399.2 Independent compound events 346

Try this: Party time 3559.3 Dependent events 3559.4 Games and other applications 362

Try this: ‘Fair’ games 369Focus on working mathematically:

Playing cricket with dice 369Language link with Macquarie 372Chapter review 372

Chapter 10 Furthertrigonometry 37710.1 Redefining the trigonometric

ratios 37810.2 Trigonometric ratios of obtuse

angles 38110.3 The sine rule 385

Try this: Double trouble 390

Mathscape 10 ext. - Prelims Page iv Tuesday, October 4, 2005 3:52 PM

v

10.4 Applications of the sine rule 39110.5 The cosine rule 395

Try this: Exact length 39910.6 Applications of the cosine rule 40010.7 Area of a triangle 404

Try this: Exact area of a segment 40810.8 Miscellaneous problems 409

Try this: Chord and radius 415Focus on working mathematically:

Measuring the distance to the stars 416Language link with Macquarie 420Chapter review 420

Chapter 11 Similarity 42311.1 Similar figures 424

Try this: Investigation 42911.2 Tests for similar triangles 430

Try this: Similar triangles 1 43711.3 Solving problems that involve

similar triangles 437Try this: Similar triangles 2 440

11.4 Proofs involving similar triangles 440Try this: Ratio in triangles 445

11.5 Area of similar figures 445Try this: Area ratio in triangles 449

11.6 Volume and surface area of similar solids 449Try this: Volume ratio 453

Focus on working mathematically:Using similar triangles to solve a

construction problem 454Language link with Macquarie 457Chapter review 457

Chapter 12 Functions andlogarithms 46012.1 Functions and relations 46112.2 Function notation 465

Try this: Minimum value 46812.3 Inverse functions 46812.4 Translating graphs of functions 47412.5 Solving simple exponential

equations 47812.6 Definition of a logarithm 48012.7 The logarithm laws 481

Try this: Logarithm relation 48512.8 Solving further exponential

equations 48612.9 The logarithmic graph 487

Try this: Log challenges 490

Focus on working mathematically:How do you measure the magnitude

of an earthquake? 490Language link with Macquarie 495Chapter review 495

Chapter 13 Curve sketchingand polynomials 49813.1 Sketching y = axn + k and

y = a(x − r)n 49913.2 The circle 501

Try this: Two circle problems 50413.3 The intersection of graphs 50413.4 Definition of a polynomial 50613.5 Addition, subtraction and

multiplication of polynomials 50913.6 The division transformation 51113.7 The remainder theorem 513

Try this: Quadratic remainder 51513.8 The factor theorem 516

Try this: Algebraic remainder 51813.9 Sketching graphs of polynomial

functions 519Try this: Quadratic factor 523

13.10 Transformations of polynomial graphs 523

Focus on working mathematically:Lawyers, a curious problem and

a family of hyperbolas 526Language link with Macquarie 529Chapter review 529

Chapter 14 Circle geometry 53114.1 Circle terminology 53214.2 Chord properties of circles 536

Try this: Circumcircle symmetry 54314.3 Angle properties of circles 54314.4 Cyclic quadrilaterals 548

Try this: Angle tangle 55314.5 Tangent properties of circles 554

Try this: Length of a median 56114.6 Further circle properties 56114.7 Deductive proofs involving

circle properties 566Focus on working mathematically:

An interesting proof of the sine ruleusing circle geometry 574


Answers 583

Mathscape 10 ext. - Prelims Page v Tuesday, October 4, 2005 3:52 PM

Mathscape 10vi

Preface

Mathscape 10 extension is a comprehensive teaching and learning resource that has been written to address the new Stage 5.1/5.2/5.3 Mathematics syllabus in NSW. Our aim was to write a book that would allow more able students to grow in confidence, to improve their understanding of Mathematics and to develop a genuine appreciation of its inherent beauty. Teachers who wish to inspire their students will find this an exciting, yet very practical resource. The text encourages a deeper exploration of mathematical ideas through substantial, well-graded exercises that consolidate students’ knowledge, understanding and skills. It also provides opportunities for students to explore the history of Mathematics and to address many practical applications in contexts that are both familiar and relevant.

From a teaching perspective, we sought to produce a book that would adhere as strictly as possible to both the content and spirit of the new syllabus. Together with Mathscape 9 extension, this book allows teachers to confidently teach the Stage 5.1/5.2/5.3 courses knowing that they are covering all of the mandatory outcomes.

Mathscape 10 extension has embedded cross-curriculum content, which will support students in achieving the broad learning outcomes defined by the Board of Studies. The content also addresses the important key competencies of the Curriculum Framework, which requires students to collect, analyse and organise information; to communicate mathematical ideas; to plan and organise activities; to work with others in groups; to use mathematical ideas and techniques; to solve problems; and to use technology.

A feature of each chapter which teachers will find both challenging and interesting for their students is the ‘Focus on working mathematically’ section. Although the processes of working mathematically are embedded throughout the book, these activities are specifically designed to provoke curiosity and deepen mathematical insight. Most begin with a motivating real-life context, such as television advertising, or the gradient of a ski run, but on occasion they begin with a purely mathematical question. (These activities can also be used for assessment purposes.)

In our view, there are many legitimate, time-proven ways to teach Mathematics successfully. However, if students are to develop a deep appreciation of the subject, they will need more than traditional methods. We believe that all students should be given the opportunity to appreciate Mathematics as an essential and relevant part of life. They need to be given the opportunity to begin a Mathematical exploration from a real-life context that is meaningful to them. To show interest and enjoyment in enquiry and the pursuit of mathematical knowledge, students need activities where they can work with others and listen to their arguments, as well as work individually. To demonstrate confidence in applying their mathematical knowledge and skills to the solution of everyday problems, they will need experience of this in the classroom. If they are to learn to persevere with difficult and challenging problems, they will need to experience these sorts of problems as well. Finally, to recognise that mathematics has been developed in many cultures in response to human needs, students will need experiences of what other cultures have achieved mathematically.

We have tried to address these values and attitudes in this series of books. Our best wishes to all teachers and students who are part of this great endeavour.

Clive MeyersLloyd DaweGraham BarnsleyLindsay Grimison

Mathscape 10 ext. - Prelims Page vi Tuesday, October 4, 2005 3:52 PM

vii

How to use this bookMathscape 10 extension is a practical resource that can be used by teachers to supplement their teaching program. The exercises in this book and the companion text (Mathscape 9 extension) provide a complete and thorough coverage of all content and skills in the Stage 5.1/5.2/5.3 course. The great number and variety of questions allow for the effective teaching of more able students. Each chapter contains:• a set of chapter outcomes directed to the student• all relevant theory and explanations, with important definitions and formulae boxed and coloured• step-by-step instructions for standard questions• a large number of fully worked examples preceding each exercise• extensive, thorough and well-graded exercises that cover each concept in detail• chapter-related, problem-solving activities called ‘Try this’ integrated throughout• a language skills section linked to the Macquarie Learners Dictionary• novel learning activities focusing on the processes of working mathematically• a thorough chapter review.

Explanations and examplesThe content and skills required to complete each exercise have been introduced in a manner and at a level that is appropriate to the students in this course. Important definitions and formulae have been boxed and coloured for easy reference. For those techniques that require a number of steps, the steps have been listed in point form, boxed and coloured. Each exercise is preceded by several fully worked examples. This should enable the average student to independently complete the majority of relevant exercises if necessary.

The exercisesThe exercises have been carefully graded into three distinct sections:• Introduction. The questions in this section are designed to introduce students to the most basic concepts

and skills associated with the outcome(s) being covered in the exercise. Students need to have mastered these ideas before attempting the questions in the next section.

• Consolidation. This is a major part of the exercise. It allows students to consolidate their understanding of the basic ideas and apply them in a variety of situations. Students may need to use content learned or skills acquired in previous exercises or topics to answer some of these questions. The average student should be able to complete most of the questions in this section, although the last few questions may be a little more difficult.

• Further applications. Some questions presented in this section will be accessible to the average student; however, the majority of questions are difficult. They might require a reverse procedure, the use of algebra, more sophisticated techniques, a proof, or simply time-consuming research. The questions can be open-ended, requiring an answer with a justification. They may also involve extension or off-syllabus material. In some questions, alternative techniques and methods of solution other than the standard method(s) may be introduced, which may confuse some students.

Teachers need to be selective in the questions they choose for their students. Some students may not need to complete all of the questions in the Introduction or Consolidations sections of each exercise, while only the most able students should usually be expected to attempt the questions in the Further applications section. Those questions not completed in class might be set as homework at the teacher’s discretion. It is not intended that any student would attempt to answer every possible question in each exercise.

Focus on working mathematicallyThe Working Mathematically strand of the syllabus requires a deeper understanding of Mathematics than do the other strands. As such, it will be the most challenging strand for students to engage with and for teachers to assess. The Working Mathematically outcomes listed in the syllabus have been carefully integrated into each chapter of the book; however, we also decided to include learning activities in each chapter that will enable teachers to focus sharply on the processes of working mathematically. Each activity begins with a

Mathscape 10 ext. - Prelims Page vii Tuesday, October 4, 2005 3:52 PM

Mathscape 10viii

real-life context and the Mathematics emerges naturally. Teachers are advised to work through them before using them in class. Answers have not been provided, but notes for teachers have been included on the Mathscape 10/10 Extension School CD-ROM, with suggested weblinks. Teachers may wish to select and use the Learning activities in ‘Focus on working mathematically’ for purposes of assessment. This too is encouraged. The Extension activities will test the brightest students. Suggestions are also provided to assess the outcomes regarding Communication and Reflection.

Problem solvingEach chapter contains a number of small, chapter-related, problem-solving activities called ‘Try this’. They may be of some historical significance, or require an area outside the classroom, or require students to conduct research, or involve the use of algebra, while others relate the chapter content to real-life context. Teachers are advised to work through these exercises before using them in class.

TechnologyThe use of technology is a clear emphasis in the new syllabus. Innovative technology for supporting the growth of understanding of mathematical ideas is provided on the Mathscape 10/10 Extension School CD-ROM, which is fully networkable and comes free-of-charge to schools adopting Mathscape 10 extension for student use. Key features of the CD-ROM include:• spreadsheet activities• dynamic geometry• animations• executables• student worksheets• weblinks for ‘Focus on working mathematically’.

LanguageThe consistent use of correct mathematical terms, symbols and conventions is emphasised strongly in this book, while being mindful of the students’ average reading age. Students will only learn to use and spell correct mathematical terms if they are required to use them frequently in appropriate contexts. A language section has also been included at the end of each chapter titled ‘Language link with Macquarie’, where students can demonstrate their understanding of important mathematical terms. This might, for example, include explaining the difference between the mathematical meaning and the everyday meaning of a word. Most chapters include a large number of worded problems. Students are challenged to read and interpret the problem, translate it into mathematical language and symbols, solve the problem, then give the answer in an appropriate context.

Clive MeyersLloyd DaweGraham BarnsleyLindsay Grimison

Mathscape 10 ext. - Prelims Page viii Tuesday, October 4, 2005 3:52 PM

mathscape 10 ext prelims

Documents