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627 Chapel Street, South Yarra 3141 Visit our website atwww.macmillan.com.au Associated companies and representativesthroughout the world. Copyright Clive Meyers, Graham Barnsley,Lloyd Dawe, Lindsay Grimison 2005 All rights reserved. Except underthe conditions described in the Copyright Act 1968 of Australia(the Act) and subsequent amendments, no part of this publicationmay be reproduced, stored in a retrieval system, or transmitted inany form or by any means, electronic, mechanical, photocopying,recording or otherwise, without the prior written permission of thecopyright owner. Educational institutions copying any part of thisbook for educational purposes under the Act must be covered by aCopyright Agency Limited (CAL) licence for educational institutionsand must have given a remuneration notice to CAL. Licencerestrictions must be adhered to. For details of the CAL licencecontact: Copyright Agency Limited, Level 19, 157 Liverpool Street,Sydney, NSW 2000. Telephone: (02) 9394 7600. Facsimile: (02) 93947601. Email: [email protected] National Library of Australiacataloguing in publication data Meyers, Clive. Mathscape 10extension. For secondary school students. ISBN 0 7329 8087 9. 1.Mathematics Textbooks. I. Grimison, Lindsay. II. Barnsley, GrahamJ. III. Dawe, Lloyd. IV. Title. 510 Publisher: Ben Dawe Projecteditor: Colin McNeil Editor: Colin McNeil Illustrator: StephenFrancis Cover and text designer: Dimitrios Frangoulis Typeset in11/13 pt Times by Sun Photoset Pty Ltd Cover image:Photolibrary.com/Science Photo Library/John Mead Printed inMalaysia Internet addresses At the time of printing, the Internetaddresses appearing in this book were correct. Owing to the dynamicnature of the Internet, however, we cannot guarantee that all theseaddresses will remain correct. Publishers acknowledgments Theauthors and publishers would like to gratefully credit oracknowledge the following for permission to reproduce copyrightmaterial: AAP, p. 277; APL/Corbis/Roger Ressmeyer, p. 242; BritishFilm Institute, p. 214; Corbis, pp. 127; Digital Stock, pp. 3268,195216, 21979, 42356; Digital Vision, p. 110; Fairfax Photos/AngelaWylie, 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. 53177, /Henry Kippert,pp. 498528; Photodisc, pp. 72113, 11859, 16492, 285331, 33971,377419, 46094; Photolibrary/Kindra Clineff, p. 190; Rob CrusePhotography, pp. 17, 25; The Art Archive, p.526. The following TryThis activities are text extracts from New Course Mathematics 10Advanced by Paul Bigelow and Graeme Stone, Macmillan EducationAustralia, 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 PaulBigelow and Graeme Stone, Macmillan Education Australia, 1998, pp.280, 444, 453; The following Language Links are text extracts fromMacquarie Learners Dictionary, Macquarie Library, 1999, pp. 28, 68,113, 160, 192, 279, 332, 372, 420, 457, 495. While every care hasbeen taken to trace and acknowledge copyright, the publisherstender their apologies for any accidental infringement wherecopyright has proved untraceable. They would be pleased to come toa suitable arrangement with the rightful owner in each case.
PrefaceMathscape 10 extension is a comprehensive teaching andlearning resource that has been written to address the new Stage5.1/5.2/5.3 Mathematics syllabus in NSW. Our aim was to write abook that would allow more able students to grow in confidence, toimprove their understanding of Mathematics and to develop a genuineappreciation of its inherent beauty. Teachers who wish to inspiretheir students will find this an exciting, yet very practicalresource. The text encourages a deeper exploration of mathematicalideas through substantial, well-graded exercises that consolidatestudents knowledge, understanding and skills. It also providesopportunities for students to explore the history of Mathematicsand to address many practical applications in contexts that areboth familiar and relevant. From a teaching perspective, we soughtto produce a book that would adhere as strictly as possible to boththe content and spirit of the new syllabus. Together with Mathscape9 extension, this book allows teachers to confidently teach theStage 5.1/5.2/5.3 courses knowing that they are covering all of themandatory outcomes. Mathscape 10 extension has embeddedcross-curriculum content, which will support students in achievingthe broad learning outcomes defined by the Board of Studies. Thecontent also addresses the important key competencies of theCurriculum Framework, which requires students to collect, analyseand organise information; to communicate mathematical ideas; toplan and organise activities; to work with others in groups; to usemathematical ideas and techniques; to solve problems; and to usetechnology. A feature of each chapter which teachers will find bothchallenging and interesting for their students is the Focus onworking mathematically section. Although the processes of workingmathematically are embedded throughout the book, these activitiesare specifically designed to provoke curiosity and deepenmathematical insight. Most begin with a motivating real-lifecontext, such as television advertising, or the gradient of a skirun, but on occasion they begin with a purely mathematicalquestion. (These activities can also be used for assessmentpurposes.) In our view, there are many legitimate, time-proven waysto teach Mathematics successfully. However, if students are todevelop a deep appreciation of the subject, they will need morethan traditional methods. We believe that all students should begiven the opportunity to appreciate Mathematics as an essential andrelevant part of life. They need to be given the opportunity tobegin a Mathematical exploration from a real-life context that ismeaningful to them. To show interest and enjoyment in enquiry andthe pursuit of mathematical knowledge, students need activitieswhere they can work with others and listen to their arguments, aswell as work individually. To demonstrate confidence in applyingtheir mathematical knowledge and skills to the solution of everydayproblems, they will need experience of this in the classroom. Ifthey are to learn to persevere with difficult and challengingproblems, they will need to experience these sorts of problems aswell. Finally, to recognise that mathematics has been developed inmany cultures in response to human needs, students will needexperiences of what other cultures have achieved mathematically. Wehave tried to address these values and attitudes in this series ofbooks. Our best wishes to all teachers and students who are part ofthis great endeavour. Clive Meyers Lloyd Dawe Graham BarnsleyLindsay Grimison
How to use this bookMathscape 10 extension is a practicalresource that can be used by teachers to supplement their teachingprogram. The exercises in this book and the companion text(Mathscape 9 extension) provide a complete and thorough coverage ofall content and skills in the Stage 5.1/5.2/5.3 course. The greatnumber and variety of questions allow for the effective teaching ofmore able students. Each chapter contains: a set of chapteroutcomes directed to the student all relevant theory andexplanations, with important definitions and formulae boxed andcoloured step-by-step instructions for standard questions a largenumber of fully worked examples preceding each exercise extensive,thorough and well-graded exercises that cover each concept indetail chapter-related, problem-solving activities called Try thisintegrated throughout a language skills section linked to theMacquarie Learners Dictionary novel learning activities focusing onthe processes of working mathematically a thorough chapterreview.
The exercisesThe exercises have been carefully graded into threedistinct sections: Introduction. The questions in this section aredesigned to introduce students to the most basic concepts andskills associated with the outcome(s) being covered in theexercise. Students need to have mastered these ideas beforeattempting the questions in the next section. Consolidation. Thisis a major part of the exercise. It allows students to consolidatetheir understanding of the basic ideas and apply them in a varietyof situations. Students may need to use content learned or skillsacquired in previous exercises or topics to answer some of thesequestions. The average student should be able to complete most ofthe questions in this section, although the last few questions maybe a little more difficult. Further applications. Some questionspresented in this section will be accessible to the averagestudent; however, the majority of questions are difficult. Theymight require a reverse procedure, the use of algebra, moresophisticated techniques, a proof, or simply time-consumingresearch. The questions can be open-ended, requiring an answer witha justification. They may also involve extension or off-syllabusmaterial. In some questions, alternative techniques and methods ofsolution other than the standard method(s) may be introduced, whichmay confuse some students. Teachers need to be selective in thequestions they choose for their students. Some students may notneed to complete all of the questions in the Introduction orConsolidations sections of each exercise, while only the most ablestudents should usually be expected to attempt the questions in theFurther applications section. Those questions not completed inclass might be set as homework at the teachers discretion. It isnot intended that any student would attempt to answer everypossible question in each exercise.
TechnologyThe use of technology is a clear emphasis in the newsyllabus. Innovative technology for supporting the growth ofunderstanding of mathematical ideas is provided on the Mathscape10/10 Extension School CD-ROM, which is fully networkable and comesfree-of-charge to schools adopting Mathscape 10 extension forstudent use. Key features of the CD-ROM include: spreadsheetactivities dynamic geometry animations executables studentworksheets weblinks for Focus on working mathematically. 153554b96e
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