Essential Mathematics for the Australian Curriculum Year 9 [Revised] 0521178657, 9780521178655

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ESSENTIAL MATHEMATICS FOR THE AUSTRALIAN CURRICULUM

9

YEAR

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DAVID GREENWOOD | SARA WOOLLEY JENNIFER VAUGHAN | JENNY GOODMAN © Greenwood et al. 2011 ISBN 978-0-521-17865-5 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

477 Williamstown Road, Port Melbourne, VIC 3207, Australia Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.edu.au Information on this title: www.cambridge.org/9780521178655 © David Greenwood, Jenny Goodman, Jennifer Vaughan, Sara Woolley, GT Installations, Georgia Sotiriou, Voula Sotiriou 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 2012 (three times), 2013, 2014 Edited by Marcia Bascombe Cover designed by Denise Lane at Sardine Designs Designed by Mary Clarke at Mason Design Typeset by Aptara Corp Printed in Singapore by C.O.S. Printers Pte Ltd A Cataloguing-in-Publication entry is available from the catalogue of the National Library of Australia at www.nla.gov.au ISBN 978-0-521-17865-5 Paperback Additional resources for this publication at www.cambridge.edu.au/GO 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) 9394 7600 Facsimile: (02) 9394 7601 Email: [email protected] 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 is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. Cover: 2011 Used under license from Shutterstock.com/ Stewie74 Images: Bernegger_Manuale/Wikimedia Commons, p. 159; NEW LINE CINEMA / THE KOBAL COLLECTION, p. 192; © Michael B. Fritz, p. 235; ©James Gurney 2006 p. 380; CDC/ Dr. Ray Butler, p. 324; NASA/courtesy of nasaimages.org/, p. 355 (bottom); All other photos: 2011 Used under license from Shutterstock.com. 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.

© Greenwood et al. 2011 ISBN 978-0-521-17865-5 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

Table of Contents Strand and substrand

About the authors Introduction and how to use this series Cambridge Go

1

Reviewing number and financial mathematics

2

Pre-test Integer operations Decimals and rounding Rational numbers Operations with fractions Ratios, rates and best buys Percentages and money Percentage increase and decrease Profits and discounts Income and taxation Simple interest Compound interest Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

4 5 10 15 20 26 32 37 42 47 54 59 63 65 66 67 68 69

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

2

viii ix xii

Linear and simultaneous equations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Pre-test Algebraic expressions Simplifying algebraic expressions Expanding algebraic expressions Solving linear equations Equations with brackets and variables on both sides Solving word problems Inequalities Using formulas Simultaneous equations: substitution Simultaneous equations: elimination

70 72 73 78 83 87 91 95 100 104 109 113

Number and Algebra -ONEYANDüNANCIALMATHEMATICS

5SINGA#!3CALCULATOR .UMBER ANDINTERESTPROBLEMSWEBSITE

Number and Algebra 0ATTERNSANDALGEBRA

5SINGA#!3CALCULATOR3OLVING EQUATIONSWEBSITE

5SINGA#!3CALCULATOR 3OLVING SIMULTANEOUSEQUATIONSWEBSITE

iii © Greenwood et al. 2011 ISBN 978-0-521-17865-5 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

2.11

3

118 122 124 125 126 127 129

Pythagoras’ theorem and trigonometry

130

Pre-test Pythagoras’ theorem Finding the shorter sides Applying Pythagoras’ theorem Pythagoras in three dimensions Trigonometric ratios Finding side lengths Solving for the denominator Finding an angle Applying trigonometry Bearings Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

132 133 139 144 149 153 159 164 168 172 178 184 186 187 188 189 191

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

4

Applications of simultaneous equations Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

Linear relations 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

Pre-test Introduction to linear relations Graphing straight lines with intercepts Lines with one intercept Gradient Gradient and direct proportion Gradient–intercept form Finding the equation of a line Midpoint and length of a line segment Perpendicular and parallel lines Linear modelling Graphical solutions to simultaneous equations Investigation Challenges

192 194 195 202 206 212 219 224 230 235 239 243 248 253 255

Measurement and Geometry 0YTHAGORASANDTRIGONOMETRY

5SINGA#!3CALCULATOR 4RIGONOMETRYWEBSITE

Number and Algebra 2EALNUMBERS ,INEARANDNON LINEARRELATIONSHIPS 5SINGA#!3CALCULATOR  3KETCHINGSTRAIGHTLINESWEBSITE

5SINGA#!3CALCULATOR&INDING INTERSECTIONPOINTSWEBSITE

iv © Greenwood et al. 2011 ISBN 978-0-521-17865-5 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

5 Measurement 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

6

Pre-test Length Circumference and perimeter of a sector Area Composite shapes Surface area of prisms and pyramids Surface area of a cylinder Volume Volume of a cylinder Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

256 257 259 261

262 264 265 270 276 284 290 295 299 305 309 310 311 312 313 315

Semester review 1

316

Indices and surds

324

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Pre-test Index notation Index laws 1 and 2 Index law 3 and the zero power Index laws 4 and 5 Negative indices Scientific notation Scientific notation using significant figures Fractional indices and surds Simple operations with surds Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

326 327 332 337 342 347 352 358 363 368 372 374 375 376 377 378

Measurement and Geometry 5SINGUNITSOFMEASUREMENT

5SINGA#!3CALCULATOR -EASUREMENTFORMULASWEBSITE

Number and Algebra 2EALNUMBERS 0ATTERNSANDALGEBRA

Measurement and Geometry 5SINGUNITSOFMEASUREMENT

v © Greenwood et al. 2011 ISBN 978-0-521-17865-5 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

7

Geometry 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

8

Pre-test Angles and triangles Parallel lines Quadrilaterals and other polygons Congruent triangles Using congruence in proof Enlargement and similar figures Similar triangles Proving and applying similar triangles Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

Algebraic techniques 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11

Pre-test Expanding binomial products Perfect squares and difference of perfect squares Factorising algebraic expressions Factorising the difference of two squares Factorisation by grouping Factorising quadratic trinomials Factorising trinomials of the form ax 2  bx  c Simplifying algebraic fractions: multiplication and division Simplifying algebraic fractions: addition and subtraction Further simplification of algebraic fractions Equations with algebraic fractions Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

380 382 383 391 398 404 411 416 423 430 435 437 438 439 440 443

444 446 447 451 456 460 463 466 470

Measurement and Geometry 'EOMETRICREASONING

Number and Algebra 0ATTERNSANDALGEBRA

5SINGA#!3CALCULATOR %XPANDINGANDFACTORISINGWEBSITE

473 478 482 486 491 493 494 495 496 497

vi © Greenwood et al. 2011 ISBN 978-0-521-17865-5 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

9

Probability and statistics 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

10

Pre-test Probability review Venn diagrams and two-way tables Using set notation Multiple events using tables Tree diagrams Experimental probability Summarising data: measures of centre Stem-and-leaf plots Grouped data Measures of spread Box plots Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

Introduction to quadratic equations and graphs 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Pre-test Quadratic equations Solving ax 2  bx  0 and x 2 d 2  0 Solving x 2  bx  c  0 Applications of quadratic equations The parabola Sketching y  ax 2 with dilations and reflections Translations of y = x 2 Sketching parabolas using intercept form Investigation Challenges Review: Chapter summary Multiple-choice questions Short-answer questions Extended-response questions

498 500 501 507 514 519 525 530 535 540 546 552 557 563 565 566 567 568 570

572 574 575 580 583 587 591 599 606 614 620 622 623 624 625 627

Semester review 2

628

Answers

635

Statistics and Probability #HANCE $ATAREPRESENTATIONANDINTERPRETATION

5SINGA#!3CALCULATOR&INDING MEASURESOFCENTREWEBSITE 5SINGA#!3CALCULATOR 'RAPHING GROUPEDDATAWEBSITE 5SINGA#!3CALCULATOR&INDING MEASURESOFSPREADANDDRAWINGBOX PLOTSWEBSITE

Number and Algebra ,INEARANDNON LINEARRELATIONSHIPS

5SINGA#!3CALCULATOR 3OLVING QUADRATICEQUATIONSWEBSITE

5SINGA#!3CALCULATOR 3KETCHINGPARABOLASWEBSITE

vii © Greenwood et al. 2011 ISBN 978-0-521-17865-5 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

About the authors David Greenwood^h=ZVYd[BVi]ZbVi^XhViIg^c^inL7

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Challenges

Chapter summary:b^cYbVed[ `ZnXdcXZeih^ciZgXdccZXi^dch 67

Rates, ratios and best buy

Operations with fractions

Rounding

5 7 = 11 – 7 16 – 12 6 12

1.284 to 1 d.p. is 1.3 2 sig. fig. is 1.3 472.543 to 1 d.p. is 472.5 1 sig. fig. is 500

= 15 12 1

=1 4 1 3 2 12 ÷ 72 = 2 × 7 1 = 37

1

180 km per 3 hrs = 60 km/h 3:1= 6 : 7 =6:7

3 11 18 = 8 = 1.375

7 2 14 14

$200 divided into 7 : 3 10 parts is $200 1 part is $20 3 parts is $60 7 parts is $140 Best buy: 3 kg carrots for $6.45 5 kg for $10.20 3 kg bag : 6.45 ÷ 3 = $2.15/kg 5 kg bag : 10.20 ÷ 5 = $2.04/kg