CambridgeMATHS NSW Stage 5 - Core and Standard Paths Year 9 PDF

9 CambridgeMATHS NSW STAGE 5  CORE AND STANDARD PATHS THIRD EDITION Stuart Palmer, Karen McDaid David Greenwood, Sara Woolley Jenny Goodman, Jennifer Vaughan  Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA...

9 CambridgeMATHS NSW STAGE 5  CORE AND STANDARD PATHS THIRD EDITION Stuart Palmer, Karen McDaid David Greenwood, Sara Woolley Jenny Goodman, Jennifer Vaughan  Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org First Edition © Stuart Palmer, David Greenwood, Jenny Goodman, Jennifer Vaughan, Sara Woolley, GT Installations, Georgia Sotiriou, Voula Sotiriou 2014 Second Edition © Stuart Palmer, Karen McDaid, David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan 2019 Third Edition © Stuart Palmer, Karen McDaid, David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan 2024 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 & Assessment. 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CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  iii Contents About the authors viii Acknowledgements x Introduction xi Guide to the working programs in exercises xiii Guide to this resource xiv 1 Integers, decimals, fractions, ratios and rates 2 Strand: Number and Algebra Warm-up quiz 4 1A Adding and subtracting positive and negative integers CONSOLIDATING 6 1B Multiplying and dividing positive and negative integers CONSOLIDATING 11 1C Decimal places and significant figures (Core) 17 1D Rational numbers and irrational numbers CONSOLIDATING 23 1E Adding and subtracting fractions CONSOLIDATING 30 Progress quiz 35 1F Multiplying and dividing fractions CONSOLIDATING 36 1G Ratios CONSOLIDATING 42 1H Rates CONSOLIDATING 47 Maths@Work: Cooks and chefs 54 Working mathematically: Painting makeover 56 Puzzles and challenges 58 Chapter summary 59 Chapter checklist with success criteria 60 Chapter review 63 2 Financial mathematics 66 Strand: Number and Algebra Warm-up quiz 68 2A Percentages, fractions and decimals CONSOLIDATING 69 2B Applying percentages CONSOLIDATING 75 2C Percentage increase and decrease CONSOLIDATING 79 2D Profits and discounts CONSOLIDATING 85 2E Income (Core) 91 Progress quiz 97 2F Taxation (Core) 98 2G Simple interest (Core) 103 2H Applications of simple interest (Core) 108 Maths@Work: Online cake-decorating business 113 Working mathematically: Saving for a holiday 115 Puzzles and challenges 117 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  iv Contents Chapter summary 118 Chapter checklist with success criteria 119 Chapter review 121 3 Expressions and equations 124 Strand: Number and Algebra Warm-up quiz 126 3A Algebraic expressions CONSOLIDATING 127 3B Adding and subtracting algebraic expressions CONSOLIDATING 135 3C Multiplying and dividing algebraic expressions CONSOLIDATING 140 3D Expanding algebraic expressions (Core) 145 3E Linear equations with a pronumeral on one side CONSOLIDATING 151 3F Linear equations involving fractions (Core) 156 Progress quiz 161 3G Linear equations involving brackets (Core) 162 3H Equations with pronumerals on both sides (Core) 166 3I Using linear equations to solve problems (Core) 171 3J Using formulas (Core) 177 Maths@Work: Plumber 183 Working mathematically: Theme parks 185 Puzzles and challenges 187 Chapter summary 188 Chapter checklist with success criteria 189 Chapter review 191 4 Right-angled triangles 194 Strand: Measurement and Warm-up quiz 196 Space 4A Exploring Pythagoras’ theorem CONSOLIDATING 197 4B Finding the length of the hypotenuse CONSOLIDATING 202 4C Finding the lengths of the shorter sides CONSOLIDATING 208 4D Using Pythagoras’ theorem to solve two-dimensional problems CONSOLIDATING 213 4E Introducing the trigonometric ratios (Core) 219 4F Finding unknown sides (Core) 226 Progress quiz 233 4G Solving for the denominator (Core) 235 4H Finding unknown angles (Core) 241 4I Using trigonometry to solve problems (Core) 247 Maths@Work: Carpenter 253 Working mathematically: Viewing angle 255 Puzzles and challenges 257 Chapter summary 258 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  Contents v Chapter checklist with success criteria 259 Chapter review 261 5 Linear relationships 266 Strand: Number and Algebra Warm-up quiz 268 5A Points and lines on the Cartesian plane (Core) 269 5B The x -intercept and y -intercept (Core) 277 5C Graphing straight lines using intercepts (Path) EXTENDING 283 5D Lines with only one intercept (Core) 288 5E Gradient from rise and run (Core) 295 5F Gradient and direct variation 304 5G Gradient–intercept form (Core) 310 Progress quiz 317 5H Finding the equation of a line using y = mx + c (Core) 319 5I Midpoint and length of a line segment from diagrams (Core) 325 5J Linear relationships in real-life contexts (Core) 331 Maths@Work: Trading in foreign currencies 336 Working mathematically: Running in a triathlon 338 Puzzles and challenges 340 Chapter summary 341 Chapter checklist with success criteria 342 Chapter review 345 Semester review 1 350 6 Length, area, surface area and volume 358 Strand: Measurement and Warm-up quiz 360 Space 6A Length and perimeter CONSOLIDATING 361 6B Circumference of circles and perimeter of sectors CONSOLIDATING 367 6C Area of quadrilaterals and triangles CONSOLIDATING 373 6D Area of circles CONSOLIDATING 380 6E Perimeter and area of composite shapes (Core) 385 6F Surface area of prisms (Core) 391 Progress quiz 397 6G Surface area of cylinders (Core) 399 6H Volume of prisms (Core) 404 6I Volume of cylinders (Core) 411 Maths@Work: Vegetable and fruit growers 416 Working mathematically: Pressing a new road 418 Puzzles and challenges 420 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  vi Contents Chapter summary 421 Chapter checklist with success criteria 422 Chapter review 424 7 Indices 428 Strand: Number and Algebra Warm-up quiz 430 7A Index notation (Core) 431 7B Index laws for multiplying and dividing (Core) 440 7C The zero index and power of a power (Core) 447 7D Index laws extended (Core) 453 7E Negative indices (Core) 460 Progress quiz 466 7F Scientific notation (Core) 467 7G Scientific notation using significant figures (Core) 472 Maths@Work: Lab technician 479 Working mathematically: Rabbits and hares 481 Puzzles and challenges 483 Chapter summary 484 Chapter checklist with success criteria 485 Chapter review 487 8 Properties of geometrical figures 490 Strand: Measurement and Warm-up quiz 492 Space 8A Angles and triangles CONSOLIDATING 493 8B Parallel lines CONSOLIDATING 504 8C Quadrilaterals CONSOLIDATING 511 8D Polygons (Path) EXTENDING 517 Progress quiz 524 8E Enlargement and similar figures (Core) 525 8F Applying scale factor to similar triangles (Core) 533 Maths@Work: Animator 538 Working mathematically: Ocean ironman 540 Puzzles and challenges 542 Chapter summary 543 Chapter checklist with success criteria 544 Chapter review 546 9 Quadratic expressions and algebraic fractions 550 Strand: Number and Algebra Warm-up quiz 552 9A Reviewing algebra CONSOLIDATING 553 9B Expanding binomial products (Core) 558 9C Expanding perfect squares (Core) 563 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  Contents vii 9D Difference of two squares (Core) 569 9E Using HCF to factorise algebraic expressions CONSOLIDATING 573 Progress quiz 578 9F Simplifying algebraic fractions: multiplication and division (Core) 579 9G Simplifying algebraic fractions: addition and subtraction (Core) 585 Maths@Work: Automotive technology 590 Working mathematically: Square castle moats 592 Puzzles and challenges 594 Chapter summary 595 Chapter checklist with success criteria 596 Chapter review 597 10 Probability and data analysis 600 Strand: Statistics and Probability Warm-up quiz 602 10A Probability review CONSOLIDATING 603 10B Venn diagrams and two-way tables (Path) EXTENDING 612 10C Using arrays for two-step experiments (Core) 622 10D Using tree diagrams (Core) 629 10E Using relative frequencies to estimate probabilities (Core) 635 10F Mean, median, mode and range CONSOLIDATING 640 Progress quiz 646 10G Interpreting data from tables and graphs (Core) 648 10H Stem-and-leaf plots (Core) 659 10I Grouping data into classes (Core) 667 Maths@Work: Personal trainer 675 Working mathematically: The missing data 677 Puzzles and challenges 679 Chapter summary 680 Chapter checklist with success criteria 681 Chapter review 683 Semester review 2 688 Answers 697 Index 773 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  viii About the authors Stuart Palmer was born and educated in NSW. He is a fully qualified high school mathematics teacher with more than 25 years’ experience teaching students from all walks of life in a variety of schools. He has been Head of Mathematics in two schools. He is well known by teachers throughout the state for the professional learning workshops he delivers. Stuart also assists thousands of Year 12 students every year as they prepare for their HSC Examinations. At the University of Sydney, Stuart spent more than a decade running tutorials for pre-service mathematics teachers. David Greenwood is the Head of Mathematics at Trinity Grammar School in Melbourne and has 25+ years’ experience teaching mathematics from Years 7 to 12. He has run numerous workshops within Australia and overseas regarding the implementation of the Australian Curriculum and the use of technology for the teaching of mathematics. He has written more than 30 mathematics titles and has a particular interest in the sequencing of curriculum content and working with the Australian Curriculum proficiency strands. Sara Woolley was born and educated in Tasmania. She completed an Honours degree in Mathematics at the University of Tasmania before completing her education training at the University of Melbourne. She has taught mathematics in Victoria from Years 7 to 12 since 2006 and is currently a Head of Mathematics. She has written more than 15 mathematics titles and specialises in lesson design and differentiation. Jennifer Vaughan has taught secondary mathematics for more than 30 years in New South Wales, Western Australia, Queensland and New Zealand, and has tutored and lectured in mathematics at Queensland University of Technology. She is passionate about providing students of all ability levels with opportunities to understand and to have success in using mathematics. She has had extensive experience in developing resources that make mathematical concepts more accessible and hence, facilitating student confidence, achievement and an enjoyment of maths. CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  About the authors ix Jenny Goodman has taught in schools for more than 25 years and is currently teaching at a selective high school in Sydney. Jenny has an interest in the importance of literacy in mathematics education, and in teaching students of differing ability levels. She was awarded the Jones Medal for Education at Sydney University and the Bourke Prize for Mathematics. She has written for CambridgeMATHS NSW and was involved in the Spectrum and Spectrum Gold series. Karen McDaid has more that 20 years’ experience teaching mathematics in primary and secondary schools, and as a lecturer teaching mathematics education to primary pre-service teachers at university. As an executive member and Past President of the Mathematics Association of NSW, she has been heavily involved in Mathematics K–10 Curriculum Consultation in NSW and across Australia. Karen co-authored the CambridgeMATHS NSW GOLD books Years 7 to 10. Karen is currently working in primary mathematics education at The Australian Catholic University. CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  x Acknowledgements The author and publisher wish to thank the following sources for permission to reproduce material: Cover: © Getty images / Merr Watson/Women Who Drone Images: © Getty images / Tom Werner, Chapter 1 Opener / Dan Reynolds Photography, p.6 / Vlatko Gasparic, p.11/ Eric O’Connell, p.16 / Nobilior, p.17 / Westend61, p.21 / sanjeri, p.22 / Busà Photography, p.29 / Annabelle Breakey, p.35 / Gaelle Beller Studio, p.36 / EujarimPhotography, p.40 / seksan Mongkhonkhamsao, p.41 / ArtMarie, p.42 / Betsie Van der Meer, p.46(1) / Halfdark, p.46(2) / Prapass, p.47(1) / Eric Isselee, p.47(2)(3)(5)(6), p.48(4)(6) / Donovan van Staden, p.47(4) / Jps, p.48(1) / Svetlana Foote, p.48(2) / Anan Kaewkhammul, p.48(3) / Aaron Amat, p.48(5) / Ryan Pierse / Staff, p.51(1) / aluxum, p.51(2) / Michael Hall, p.52 / Image Source, p.53(1) / Milo Zanecchia/ Ascent Xmedia, p.53(2) / Westend61, p.54(1) / Nina Van Der Kleij / EyeEm, p.54(2) / La Bicicleta Vermella, p.55 / 10’000 Hours, p.56 / andresr, p.57 / peepo, p.63 / Jacobs Stock Photography Ltd, p.65 / Andrew Aitchison / Contributor, Chapter 2 Opener / Kentaroo Tryman, p.73 / Waldemar Blazej Nowak / EyeEm, p.74(1) / Jose Luis Pelaez Inc, p.74(2) / skynesher, p.75 / Julian Elliott Photography, p.78 / yasinguneysu, p.83 / clubfoot, p.87 / studiocasper, p.88 / JazzIRT, p.90(1) / Bruce Yuanyue Bi, p.90(2) / Morsa Images, p.96 / Tom Werner, p.102 / Chris Ryan, p.102 / Firdausiah Mamat, p.105 / Betsie Van der Meer, p.107 / urbazon, p.108 / JGI/Jamie Grill, p.111 / Aleksandr Zubkov, p.114 / courtneyk, p.121 / Stijn Dijkstra / EyeEm, Chapter 3 Opener / Halfdark, p.130 / JohnnyGreig, p.133(1) / Yulia Naumenko, p.133(2) / Kanok Sulaiman, p.134 / Dawid Zebrowski, p.135 / NeilyImagery, p.139 / Westend61, p.149 / 10’000 Hours, p.150 / rolfo eclaire, p.160 / JohnnyGreig, p.165 / Arno Images, p.173 / Cavan Images, p.175(1) / Tom Werner, p.175(2) / ivanastar, p.176 / Boy_Anupong, p.177 / Emanuel M Schwermer, p.181 / Image Source, p.183 / apomares, p.186 / Lindy Kyle / EyeEm, p.187 / MediaProduction, p.193 / Dan Reynolds Photography, Chapter 4 Opener / ZU_09, p.197 / roman_slavik, p.202 / courtneyk, p.207 / Fotosearch, p.208 / Robin Smith, p.213 / koiguo, p.218(1) / Tommy Colombet / EyeEm, p.218(2) / Ricardo Liberato, p.219 / Image Source, p.224 / Luca Sgualdini, p.231 / Huntstock, p.234 / Tommy Colombet / EyeEm, p.235 / Frisco, p.241 / gilaxia, p.245 / Atlantide Phototravel, p.246 / Bim, p.250 / Luca Sgualdini, p.251(1) / Simon McGill, p.251(2) / Daniel H. Bailey, p.252 / Guido Mieth, p.253 / Matt Carr, p.256 / Vostok, p.261 / Mads Wollesen / EyeEm, p.265 / The Creative Drone, Chapter 5 Opener / darrya, p.275 / Virojt Changyencham, p.277 / Darya Bystritskaya / EyeEm, p.282 / Andriy Onufriyenko, p.283 / Trevor Watchous / EyeEm, p.287 / Image Source, p.293 / Buena Vista Images, p.295 / Imaginima, p.302 / prognone, p.308 / Suneet Bhardwaj / 500px, p.309 / Buena Vista Images, p.310 / Travis Wolfe / EyeEm, p.316 / Chris Rogers, p.328 / John Coletti, p.329 / YvanDube, p.334 / This is a Lukerative Image, p.335 / Travis Wolfe / EyeEm, p.337 / Alistair Berg, p.340 / Glasshouse Images, p.342 / Kyle Willis, p.346 / carlosalvarez, p.349 / Abstract Aerial Art, p.354 / Juanmonino, Chapter 6 Opener / Stocktrek Images, p.361 / Hisham Ibrahim, p.366(1) / JamesBrey, p.366(2) / xavierarnau, p.367 / Aleli Dezmen, p.371 / Cavan Images, p.372 / Achim Thomae, p.373 / Matt Champlin, p.378 / Peter Zelei Images, p.379 / Thanpisit Thongdam / EyeEm, p.384(1) / Michael Dunning, p.391 / Kitti Boonnitrod, p.396 / JustinBlackStock, 398 / Colinda Jansen / EyeEm, p.403(1) / tucko019, p.403(2) / Colorblind Images LLC, p.412 / Douglas Sacha, p.415 / Mint Images - Tim Robbins, p.416(1) / Zelma Brezinska / EyeEm, p.420 / Mr.samarn Plubkilang / EyeEm, p.427 / Pekic, Chapter 7 Opener / Rodolfo Parulan Jr, p.439 / Westend61, p.446 / vandervelden, p.452 / Barbara Rich, p.459, p.467 / LEONELLO CALVETTI, p.471(1) / KTSDESIGN/SCIENCE PHOTO LIBRARY, p.471(2) / Matthias Kulka, p.472 / Natee Meepian / EyeEm, p.475 / Stocktrek, p.477 / ROGER HARRIS/SCIENCE PHOTO LIBRARY, p.478 / Thomas Tolstrup, p.479 / vandervelden, p.482 / Roman Becker / EyeEm, Chapter 8 Opener / Tomatoskin, p.493 / hh5800, p.509 / Dan Reynolds Photography, p.510 / Nora Carol Photography, p.536 / fotostorm, p.538 / Enzo D., p.541 / andresr, Chapter 9 Opener / Matelly, p.557 / Mint Images, p.558 / JohnnyGreig, p.562 / Monty Rakusen, p.566 / Primeimages, p.572 / Lingbeek, p.593 / petesphotography, Chapter 10 Opener / Zero Creatives, p.603 / Kinga Krzeminska, p.611 / Jackyenjoyphotography, p.613 / itsabreeze photography, p.620 / Morsa Images, p.621 / Tetra Images, p.622 / Tom Werner, p.640 / Westend61, p.644 / Oliver Strewe, p.647 / Robert Lang Photography, p.655 / Martin Barraud, p.658 / SolStock, p.659 / Sam Edwards, p.661 / Hans Neleman, p.666 / taffpix, p.647 / Tommi Kokkola Photography, p.677 / Image Source, p.678 / Kiyoshi Hijiki, p.687 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. Mathematics K-10 Syllabus © NSW Education Standards Authority for and on behalf of the Crown in right of the State of New South Wales, 2022. The NSW Education Standards Authority (NESA) does not endorse model answers prepared by or for the publisher and takes no responsibility for errors in the reproduction of the material supplied by the NESA. The publisher wishes to acknowledge that this study guide has been independently published by Cambridge University Press for use by teachers and students, and that although material contained has been reproduced with permission, this publication is in no way endorsed by the NESA. CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  xi Introduction The third edition of CambridgeMATHS NSW Stages 4 and 5 has been carefully prepared for the NSW Syllabus implemented from 2024 onwards. The series and is packed with new features. It is intended for students across a wide range of ability levels and has been designed to provide the best possible preparation for students as they make their way from the end of Stage 3 towards Stage 6 mathematics. Coverage of Core and optional Path topics in Stage 5 As with the previous editions, Stage 5 is covered with two differentiated pairs of books: The Core & Standard Paths books replace the old Stage 5.1/5.2 books, The Core & Advanced/Extension Paths books replace the Stage 5.1/5.2/5.3 books. Care has been taken to choose the content in each book carefully and to present it in a way that reflects the spirit of the new syllabus as well as the practicalities of the classroom. ‘Paths’, in the plural, refers to the fact that the Path topics are optional, and that a student’s journey through Stage 5 may incorporate the core content in combination with none, some or all the Path topics in the books, depending upon the student’s needs. Detailed teaching programs and Scope and Sequence documents are available via the Online Teaching Suite. Beyond Stage 5 Students who have used these Core & Standard Paths books in Years 9 and 10 will be thoroughly prepared for Year 11 Mathematics Standard. These books are not recommended for students who have aspirations of studying Mathematics Advanced in Stage 6. Learning Intentions, Success Criteria checklist, and ‘Past, present and future learning’ Every lesson now includes a set of Learning Intentions that describe what the student can expect to learn in the lesson. At the end of the chapter, these appear again in the form of a checklist of Success Criteria; students can use this to check their progress through the chapter. Every criterion is listed with an example question to remind students of what the mathematics mentioned looks like. These checklists can also be downloaded and printed so that students can physically check them off as they accomplish their goals. Also included at the beginning of each section, under the Learning Intentions, is a short set of dot points that place the topic at hand in the context of past, present and future learning. Now you try Every worked example now contains additional questions, without solutions, called ‘Now you try’, which give students immediate practice at the same type of question as the example. We also anticipate these questions will be useful for the teacher to do in front of the class, given that students will not have seen the solution beforehand. CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  xii Introduction Working Mathematically tasks A Working Mathematically activity now accompanies an Investigation in each chapter, with the goal of familiarising students with using the Working Mathematically process to define, solve, verify and then communicate their solutions when solving non-routine problems. Maths@Work An activity called ‘Maths@Work’ is now included at the end of every chapter. The intention of these activities is to show students how the maths learnt in the chapter is relevant to the workplace in a range of industries. The first part of the activity involves students doing the work by hand, while the second part involves the use of technology – often spreadsheets – to allow for greater efficiency, power and accuracy. CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  xiii Guide to the working programs in exercises The suggested working programs in the exercises in the Year 9 and 10 Core & Standard Paths books provide two pathways to allow differentiation for Building and Progressing students (schools will have their own names for different ability levels or streams). Each exercise is structured in subsections that match the Building Progressing working mathematically components of Understanding, UNDERSTANDING 1–4 4 Fluency, Problem-solving and Reasoning, as well as FLUENCY Enrichment (Challenge). In the exercises, the questions 5–7(½) 5–8(½) suggested for each pathway are listed in two columns at the PROBLEM-SOLVING AND REASONING top of each subsection: 9–10 10–12 The left column (lightest shaded colour) is the Building ENRICHMENT: The missing y-intercept pathway – 13 The right column (darkest shaded colour) is the The working program for Exercise 5G in Year 9. The questions recommended for a Building Progressing pathway. student are: 1, 2, 3, 4, 5, 6, 7(½), 9 and 10. Gradients within exercises and proficiency strands The working programs make use of the two difficulty gradients contained within exercises. A gradient runs through the overall structure of each exercise – where there is an increasing level of mathematical sophistication required from Understanding to Fluency through to Problem-solving and Reasoning, and Enrichment – but also within each Working Mathematically component; the first few questions in Fluency, for example, are easier than the last Fluency question. The right mix of questions Questions in the working programs are selected to give the most appropriate mix of types of questions for each learning pathway. Students going through the Building pathway should use the left tab, which includes all but the hardest Understanding and Fluency questions as well as the easiest Problem-solving and Reasoning questions. The Progressing pathway, while not challenging, spends a little less time on basic Understanding questions and a little more time on Fluency and Problem-solving and Reasoning questions. Choosing a pathway ∗ The nomenclature used to list There are a variety of ways to determine the appropriate pathway for questions is as follows: students through the course. Schools and individual teachers should 3, 4: complete all parts of follow the method that works for them. If required, the Warm-up quiz at questions 3 and 4 the start of each chapter can be used as a tool for helping students select a 1–4: complete all parts of questions 1, 2, 3 and 4 pathway. The following are recommended guidelines: 10(½): complete half of the A student who gets 40% or lower should heavily revise core concepts parts from question 10 (a, c, e,..... or b, d, f,.....) before attempting the Buildings questions. 2–4(½): complete half of the A student who gets above 40% and below 75% should complete the parts of questions 2, 3 and 4 Building questions 4(½), 5: complete half of the parts of question 4 and all A student who gets 75% or higher should complete the Progressing parts of question 5 questions. — : complete none of the questions in this section. CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  xiv Guide to this resource PRINT TEXTBOOK FEATURES 1 NSW Syllabus: content strands, outcome groups and outcomes are listed at the beginning of the chapter to assist with course planning (see teaching programs and Syllabus mapping grids for more detailed guidance) 2 Chapter introduction: sets context for students about how the topic connects with the real world and the history of mathematics 3 Warm-up quiz: a quiz for students on the prior knowledge and essential skills required before beginning the chapter NEW 4 Learning intentions: sets out what a student will be expected to learn in the lesson NEW 5 Past, present and future learning: shows how the lesson builds on previous or extends to future lessons in the context of the syllabus 6 Lesson starter: an activity, which can often be done in groups, to start the lesson 7 Key ideas: summarises the knowledge and skills for the lesson 8 Worked examples: solutions and explanations of each line of working, along with a description that clearly describes the mathematics covered by the example NEW 9 Now you try: try-it-yourself questions provided after every worked example in exactly the same style as the worked example to give immediate practice 4 5 10, 11 6 7 8 9 10 Gentle start to exercises: the exercise begins at Understanding to ensure comprehension of the lesson’s Key Ideas before progressing to the other Working Mathematically components 11 Working programs: differentiated question sets for two ability levels in exercises CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  Guide to this resource xv NEW 12 Progress quiz about two-thirds of the way through each chapter provides a check-in to establish how students are going with the topic so far 13 NEW 13 Maths@Work: a set of questions that gives practice at applying the mathematics of the chapter to real-life contexts NEW 14 Working Mathematically investigative tasks: in each chapter that apply the Working Mathematically process to the topic to give students guided practice 15 Puzzles and challenges: in each chapter provide problem-solving practice in the context of puzzles and challenges connected with the topic NEW 16 Chapter checklist with success criteria: a checklist of the learning intentions for the chapter, with example questions 17 Chapter reviews: with short-answer, multiple-choice and extended-response questions; questions that are extension are clearly signposted 16 Chapter review 487 338 Chapter 5 Linear relationships Working mathematically 339 Short-answer questions Chapter review 1 Express each of the following in index form. Running in a triathlon d Levi starts running after Jack had started. Levi runs at 3.5 m/s and the rule describing Levi’s 7A a 3×3×3×3 b 2×x×x×x×y×y Working mathematically distance d after t seconds is given by d = 3.5t − 175. b 3 3 3 1 1 c 3×a×a×a× ×b d × × × × i Let d = t, Levi’s starting time. Plot this point and label its coordinates. a 5 5 5 7 7 ii Find Levi’s distance when t = 350. Plot this point and label its coordinates. iii Draw a line through these points extend it to 1200 m. Label this line ‘Levi’. 7A 2 Write the following as a product of prime factors in index form. and Max runs at 4 m/s. However, Max and Levi take longer to complete the swimming and cycling and a 45 b 300 e What occurs on the graph at t = 350? Explain what happens between Jack and Levi at 350 s. start the run after Jack started. Can either of them overtake Jack and win the race? f Max starts the running race behind Levi and Jack. Max runs at 4 m/s and the equation describing 3 Simplify using index laws. For this task, d metres is the distance travelled by a runner and t seconds is the time in seconds after 7B Max’s distance d after t seconds is given by d = 4t − 300. a x3 × x 7 b 2a3 b × 6a2 b5 c c 3m2 n × 8m5 n3 Jack starts running. i Let d = t, Max’s starting time. Plot this point and label its coordinates. 5a8 b3 d a12 ÷ a3 e x5 y3 ÷ (x2 y) f Present a report for the following tasks and ensure that you show clear mathematical workings, ii Find Max’s distance when t = 250. Plot this point and label its coordinates. 10a6 b explanations and diagrams where appropriate. iii Rule a line through these points and extend it to 1200 m. Label this line ‘Max’. g What occurs on the graph at t = 250? Explain what happens between Max and Levi at 7C/7D 4 Simplify: Routine problems 3 a Find the time, in seconds, taken to run 1200 m at an average speed of 2.4 m/s. b Write the equation for distance d after t seconds when running at a speed of 2.4 m/s. 250 seconds. h Does Max also overtake Jack? If so, state the time and distance where this happens. a (m2 )3 b (3a4 )2 c (−2a2 b)5 d 3a0 b e 2(3m)0 f ()a2 3 c An equation describing a different runner is given by d = 2.5t − 20. Let d = 1200 and solve the i 7E 5 Express each of the following with positive indices. j Let d = 1200 in each equation and solve each for t 1 6 2 Communicate a 2−3 b 3 × 4−5 c 6 × 10−3 d e f nearest second. thinking 4−2 5−3 10−5 Non-routine problems k and reasoning Explore and third runner? 7B/C/D 6 Use index laws to simplify each of the following. connect a Write down all the relevant information that will help solve this problem. k5 × k3 9a4 4m4 l a b × c (a2 )3 × 4a4 b Using a large sheet of graph paper or digital graphing software, prepare axes like those shown k2 2m3 3a2 below. Draw and label a horizontal axis showing time t from 0 seconds to 400 seconds and Extension problem 4(a3 )2 (b4 )3 d (h12 )3 ÷ (h4 )5 e f 5(3p2 q)2 a vertical axis showing distance d from 0 m to 1200 m. 20a0 (b2 )5 Problem-solve Running in a triathlon 1200 a Answer the following questions about the line on the graph that would show Elijah’s run. 7F 7 Write each of the following numbers as a basic numeral to arrange in ascending order. i State the coordinates of the start and end points that would be on Elijah’s line. 2.35, 0.007 × 102 , 0.0012, 3.22 × 10−1 , 0.4, 35.4 × 10−3. 14 1000 ii Calculate the gradient of the line that joins these two points. 17 8 Write each of the following numbers as a basic numeral. Distance d in m 800 iii What feature of Elijah’s running does this gradient represent? 7F a 3.24 × 102 b 1.725 × 105 c 2.753 × 10−1 d 1.49 × 10−3 600 iv The line equation is in the form d = mt + c. Find the values of m and c and hence the equation 400 of the line that would represent Elijah’s run. 9 7G 200 a The population of Australia in 2010 was approximately 22 475 056. b The area of the USA is 9 629 091 km2. 0 0 50 100 150 200 250 300 350 400 c The time taken for light to travel 1 metre (in a vacuum) is 0.00000000333564 seconds. Time t in seconds (after Jack starts) d The wavelength of ultraviolet light is 0.000000294 m. c Jack runs at an average speed of 3 m/s for the 1200 m. 7F 10 Choose and apply i How long, in seconds, does Jack take to run the 1200 m? Note: 1 tonne = 1000 kg. techniques ii Write an equation for Jack’s distance (d metres) in terms of time (t seconds). a 25 years (hours) b 3 million years (months) iii Draw a line on the graph to represent Jack’s run. Label this line ‘Jack’. c 430 tonnes (kg) d 5 tonnes (grams) CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  xvi Guide to this resource INTERACTIVE TEXTBOOK FEATURES 18 NEW Targeted Skillsheets, one for each lesson, focus on a small set of related Fluency-style skills for students who need extra support, with questions linked to worked examples 19 Workspaces: almost every textbook question – including all working-out – can be completed inside the Interactive Textbook by using either a stylus, a keyboard and symbol palette, or uploading an image of the work 20 Self-assessment: students can then self-assess their own work and send alerts to the teacher. See the Introduction on page xi for more information. 21 Interactive working programs can be clicked on so that only questions included in that working program are shown on the screen 22 HOTmaths resources: a huge catered library 24 26 of widgets, HOTsheets and walkthroughs 23 seamlessly blended with the digital textbook 23 A revised set of differentiated auto-marked practice quizzes per lesson with saved scores 22 24 Scorcher: the popular competitive game 25 Worked example videos: every worked example is linked to a high-quality video demonstration, supporting both in-class learning and the flipped classroom 26 Desmos graphing calculator, scientific calculator and geometry tool are always available to open within every lesson 27 Desmos interactives: a set of Desmos activities written by the authors allow students to explore a key mathematical concept by using the Desmos graphing calculator or geometry tool 25 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  Guide to this resource xvii 28 Auto-marked warm-up quiz for testing the knowledge that students will need before starting the chapter 29 Auto-marked progress quizzes and chapter review multiple-choice questions in the chapter reviews can now be completed online DOWNLOADABLE PDF TEXTBOOK 30 In addition to the Interactive Textbook, a PDF version of the textbook has been retained for times when users cannot go online. PDF search and commenting tools are enabled. 30 ONLINE TEACHING SUITE NEW 31 Diagnostic Assessment Tool included with the Online Teaching Suite allows for flexible diagnostic testing, reporting and recommendations for follow-up work to assist you to help your students to improve NEW 32 PowerPoint lesson summaries contain the main elements of each lesson in a form that can be annotated and projected in front of class 33 Learning Management System with class and student analytics, including reports and communication tools 34 Teacher view of student’s work and self-assessment allows the teacher to see their class’s workout, how students in the class assessed their own work, and any ‘red flags’ that the class has submitted to the teacher 35 Powerful test generator with a huge bank of levelled questions as well as ready-made tests 36 Revamped task manager allows teachers to incorporate many of the activities and tools listed above into teacher-controlled learning pathways that can be built for individual students, groups of students and whole classes CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  xviii Guide to this resource 37 Worksheets and two differentiated chapter tests in every chapter, provided in editable Word documents 38 More printable resources: all warm-up quizzes, Progress quizzes and Working Mathematically tasks are provided in printable worksheet versions 33 35 37 38 34 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  1 Integers, decimals, fractions, ratios and rates Maths in context: Number skills in cooking, cycling and manufacturing Number skills are essential for trades, professions and Fishermen and power boat owners mix petrol and many practical tasks. Negative numbers are used for oil in a ratio of 50 : 1 for outboard motor fuel. temperatures and in financial calculations. Skills with Jewellers can mix gold, copper and silver in a ratio rates, ratios and fractions are applied in numerous of 15 : 4 : 1 for a rose gold ring, necklace or occupations, for example: bracelet. Builders mix cement, sand and gravel in a ratio of Cooks and chefs regularly make calculations using 1 : 2 : 4 for driveway concrete and they mix cement rates, fractions and ratios when adapting recipes. and sand in a ratio of 1 : 3 for swimming pool Competitive cyclists select gear ratios to allow the concrete. maximum possible acceleration and speed for various race conditions. CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  Chapter contents 1A Adding and subtracting positive and negative integers (CONSOLIDATING) 1B Multiplying and dividing positive and negative integers (CONSOLIDATING) 1C Decimal places and significant figures 1D Rational numbers and irrational numbers (CONSOLIDATING) 1E Adding and subtracting fractions (CONSOLIDATING) 1F Multiplying and dividing fractions (CONSOLIDATING) 1G Ratios (CONSOLIDATING) 1H Rates (CONSOLIDATING) NSW Syllabus In this chapter, a student: develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) compares, orders and calculates with integers to solve problems (MA4-INT-C-01) represents and operates with fractions, decimals and percentages to solve problems (MA4-FRC-C-01) solves problems involving ratios and rates, and analyses distance–time graphs (MA4-RAT-C-01). © 2022 NSW Education Standards Authority Online resources Rate calculations are essential for farmers to A host of additional online resources are included efficiently manage water usage. Rates include pump as part of your Interactive Textbook, including flow rates (litres per second); drip irrigation rates HOTmaths content, video demonstrations of all (litres per hour); travelling irrigator rates (acres per worked examples, auto-marked quizzes and much hour); and irrigation frequency rates (number of times more. irrigation occurs per week). CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  4 Chapter 1 Integers, decimals, fractions, ratios and rates 1 Arrange the following mathematical terms under four headings: ‘Addition’, ‘Subtraction’, ‘Multiplication’ and ‘Division’. Warm-up quiz a Sum b Total c Less than d Lots of e Product f Into g Take away h Difference i Add j Times k Minus l More than m Quotient n Less o Share p Increase 2 Without using a calculator, find an answer to each of the following. a 16 less 12 b 24 more than 8 c the difference between 12 and 8 d increase 45 by 7 e the total of 40, 34 and 0 f 9 into 45 g the quotient of 63 and 7 h 480 shared between 12 3 Evaluate the following. a 9 + 47 b 135 − 35 c 19 − 19 d 56 + 89 − 12 e 9×7 f 320 ÷ 4 g 17 × 60 h 200 − 47 − 100 4 Use a number line to find: a −5 − 7 b 12 − 15 c −6 + 9 d −12 + 12 e 16 − 17 f −4 + 3 g −7 + 4 + 3 h −4 − 4 − 4 5 Copy and complete each of the following statements. a 5+5+5= ×5 b −6 − 6 − 6 = × (−6) c 9 − (+10) = 9 10 d 12 − (−2) = 12 2 6 The population of Australia in 2050 is projected to be 26 073 258. Round this number to the nearest: a ten b hundred c thousand d million 7 Write down the place value of the 5 in each of the following numbers. a 1256 b 345 c 5049 d 0.56 e 0.15 f 9.005 8 Arrange the numbers in each of the following sets in descending order (largest to smallest). a 2.645, 2.654, 2.465 and 2.564 b 0.456, 0.564, 0.0456 and 0.654 9 Evaluate each of the following. a 4.26 + 3.73 b 3.12 + 6.99 c 10.89 − 3.78 10 Evaluate: a 7 × 0.2 b 0.3 × 0.2 c 2.3 × 1.6 d 4.2 × 3.9 e 14.8 ÷ 4 f 12.6 ÷ 0.07 11 Evaluate each of the following. a 0.345 × 100 b 3.74 × 100 000 c 37.54 ÷ 1000 d 3.7754 ÷ 100 000 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  Warm-up quiz 5 12 Complete these equivalent fractions. 1 3 5 25 2 Warm-up quiz a = b = c = d = 2 12 4 16 6 9 18 13 Find the lowest common denominator for these pairs of fractions. 1 1 1 1 1 1 a and b and c and 3 5 6 4 5 10 14 Find: 3 2 3 1 3 1 a + b 2− c 4÷ d × 7 7 4 2 4 2 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  6 Chapter 1 Integers, decimals, fractions, ratios and rates 1A Adding and subtracting positive and negative integers CONSOLIDATING Learning intentions for this section: To review the use of the number line for addition and subtraction To evaluate expressions involving addition and subtraction of integers Past, present and future learning: This section consolidates Stage 4 concepts that are used in Stages 5 and 6 Some of these questions are more challenging than those in our Stage 4 books In Chapter 3 of this book, we apply this learning to algebraic expressions Expertise with integers is assumed knowledge for Stage 6 Standard and may also be required in non-calculator examinations such as NAPLAN and industry aptitude tests Integers are the set of positive and negative whole numbers, as well as the number zero. Being able to work with whole numbers is very important, since whole numbers are used every day for counting, ordering, calculating, measuring and computing. Lesson starter: Naming groups Here are some groups of numbers. In groups of two or three, use the correct mathematical terms to describe each group. (Suggestions include ‘multiples of’, ‘factors of’, ‘integers’, ‘squares’ and ‘cubes’.) 2, 4, 6, 8, … 1, 4, 9, 16, … 1, 3, 5, 7, 9, … 1, 2, 3, 4, 5, 6, … −1, −2, −3, −4, −5, … 1, 8, 27, 64, … 1, 2, 3, 4, 6 and 12 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  1A Adding and subtracting positive and negative integers 7 KEY IDEAS Integers are the positive and negative whole numbers, including zero: … , −3, −2, −1, 0, 1, 2, 3, … To add a negative number, you subtract its opposite: a + (−b) = a − b e.g. 5 + (−7) = 5 − 7 −6 + (−2) = −6 − 2 To subtract a negative number, you add its opposite: a − (−b) = a + b e.g. 5 − (−7) = 5 + 7 −6 − (−2) = −6 + 2 Adding or subtracting zero leaves a number unchanged. a+0=a e.g. 5 + 0 = 5 a−0=a e.g. 5 − 0 = 5 Key terminology: integer, positive, negative Exercise 1A UNDERSTANDING 1–3 3 1 Match each sentence to the correct expression. a The sum of 5 and 7 i 5 + (−7) b The total of negative 5 and 7 ii 5 − (−7) c The difference between negative 5 and 7 iii 5+7 d The sum of 5 and negative 7 iv 7 − (−5) e The difference between 5 and negative 7 v −5 + 7 2 Match each question on the left to an expression or answer on the right. a 10 + (−7) i −10 − 7 b −10 − (−7) ii −10 + 7 c 10 − (−7) iii 0 d −10 + (−7) iv 10 − 7 e −10 + 10 v 10 + 7 3 Are the following true (T) or false (F)? a 18 + 0 = 18 b 6 − (−4) = 6 + 4 c 4 − (−2) = 4 − 2 d 9+8=8+9 e 6 − (−4) = 6 + 4 f 4 − (−2) = 4 − 2 CambridgeMATHS NSW Stage 5 – Core and Standard Paths ISBN 978-1-009-40926-1 © Palmer et al. 2024 Cambridge University Press Year 9 Photocopying is restricted under law and this material must not be transferred to another party.  8 Chapter 1 Integers, decimals, fractions, ratios and rates FLUENCY 4–8(½) 5–8(½) Example 1 Using a number line for addition and subtraction of integers Use a number line to find: a −10 + 8 b −3 − 5 SOLUTION EXPLANATION a